NCERT Solutions for Class 12 Computer Science (C++) – Boolean Algebra

<div class=”entry-content” style=”height: auto !important;”><div id=”toc_container” class=”no_bullets contracted” style=”width: auto; display: table;”>Contents [<a href=”#”>show</a>]<ul class=”toc_list” style=”display: none;”><li><a href=”#NCERT_Solutions_for_Class_12_Computer_Science_C_8211_Boolean_Algebra”>1 NCERT Solutions for Class 12 Computer Science (C++) – Boolean Algebra</a><ul><li><a href=”#TOPIC-1_Basics_of_Boolean_Algebra”>1.1 TOPIC-1
Basics of Boolean Algebra</a></li><li><a href=”#TOPIC-2_Karnaugh_Map_Minimization_and_Applications_of_Boolean_Algebra”>1.2 TOPIC-2
Karnaugh Map Minimization and Applications of Boolean Algebra</a></li></ul></li></ul></div>
<h2>NCERT Solutions for Class 12 Computer Science (C++) – Boolean Algebra</h2>
<h3 style=”text-align: center;”>TOPIC-1 
 Basics of Boolean Algebra</h3>
Very Short Answer Type Questions    [1 mark each]<div class=”code-block code-block-1″ style=”margin: 8px 0; clear: both;”>
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Question 1: 
Which gates are known as universal gates ? Why? 
Аnswer: 
Universal gates are the ones which can be used for implementing any gate like AND, OR and NOT or any combination of these basic gates. NAND and NOR gates are universal gates.
Check the <a href=”https://www.learncram.com/calculator/boolean-algebra-calculator/”>boolean algebra calculator</a>, Look at How easily to solve Boolean Expressions.<div class=”google-auto-placed ap_container” style=”width: 100%; height: auto; clear: both; text-align: center;”><ins data-ad-format=”auto” class=”adsbygoogle adsbygoogle-noablate” data-ad-client=”ca-pub-7398766921532682″ data-adsbygoogle-status=”done” style=”display: block; margin: auto; background-color: transparent; height: 280px;” data-ad-status=”filled”><div id=”aswift_3_host” style=”border: none; height: 280px; width: 750px; margin: 0px; padding: 0px; position: relative; visibility: visible; background-color: transparent; display: inline-block; overflow: visible;” tabindex=”0″ title=”Advertisement” aria-label=”Advertisement”><iframe id=”aswift_3″ name=”aswift_3″ style=”left:0;position:absolute;top:0;border:0;width:750px;height:280px;” sandbox=”allow-forms allow-popups allow-popups-to-escape-sandbox allow-same-origin allow-scripts allow-top-navigation-by-user-activation” width=”750″ height=”280″ frameborder=”0″ marginwidth=”0″ marginheight=”0″ vspace=”0″ hspace=”0″ allowtransparency=”true” scrolling=”no” src=”https://googleads.g.doubleclick.net/pagead/ads?client=ca-pub-7398766921532682&output=html&h=280&adk=2523109437&adf=1727117068&pi=t.aa~a.3071249423~i.11~rp.4&w=750&fwrn=4&fwrnh=100&lmt=1670938749&num_ads=1&rafmt=1&armr=3&sem=mc&pwprc=6908628465&ad_type=text_image&format=750×280&url=https%3A%2F%2Fwww.cbsetuts.com%2Fncert-solutions-class-12-computer-science-c-boolean-algebra%2F&fwr=0&pra=3&rh=188&rw=750&rpe=1&resp_fmts=3&wgl=1&fa=27&adsid=ChEIgPvEnQYQ7oKUkIGo7bvHARI9AMmKl4lSlmxSp_EuqkkvujLyJB4s6dU2LtSRXWDmFUOke_SQLWbIo9A98U6dlFYFIStmRtcRp8siPDvC_w&uach=WyJXaW5kb3dzIiwiMTAuMC4wIiwieDg2IiwiIiwiMTA4LjAuNTM1OS4xMjUiLFtdLGZhbHNlLG51bGwsIjY0IixbWyJOb3Q_QV9CcmFuZCIsIjguMC4wLjAiXSxbIkNocm9taXVtIiwiMTA4LjAuNTM1OS4xMjUiXSxbIkdvb2dsZSBDaHJvbWUiLCIxMDguMC41MzU5LjEyNSJdXSxmYWxzZV0.&dt=1672590903988&bpp=4&bdt=2098&idt=-M&shv=r20221207&mjsv=m202212010101&ptt=9&saldr=aa&abxe=1&cookie=ID%3D70877dca39d7cee1-22c4db3411d900f8%3AT%3D1672348851%3ART%3D1672348851%3AS%3DALNI_MYJDyG_bu32YHEUMIGJIo2bVt3_WQ&gpic=UID%3D00000b9a5c88ec58%3AT%3D1672348851%3ART%3D1672575613%3AS%3DALNI_Mb87E1-6Z_pG79EP2w2fRKpjdy0KA&prev_fmts=0x0%2C300x600&nras=2&correlator=6454028235071&frm=20&pv=1&ga_vid=540223204.1672348852&ga_sid=1672590903&ga_hid=613024059&ga_fc=1&u_tz=330&u_his=5&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=24&u_sd=1&dmc=4&adx=105&ady=815&biw=1349&bih=657&scr_x=0&scr_y=0&eid=44759875%2C44759926%2C44759842%2C31071167%2C44779793%2C44780792%2C21065724&oid=2&pvsid=972740684238546&tmod=365514175&uas=3&nvt=1&ref=https%3A%2F%2Fwww.learncbse.in%2F&eae=0&fc=1408&brdim=0%2C0%2C0%2C0%2C1366%2C0%2C1366%2C728%2C1366%2C657&vis=1&rsz=%7C%7Cs%7C&abl=NS&fu=128&bc=31&jar=2022-12-31-10&ifi=4&uci=a!4&btvi=2&fsb=1&xpc=V5o1jJqZhI&p=https%3A//www.cbsetuts.com&dtd=135″ data-google-container-id=”a!4″ data-google-query-id=”CILkltXmpvwCFdSI6QUdvmQN8Q” data-load-complete=”true”></iframe></div></ins></div>
Question 2: 
Draw the equivalent logic circuit for the following Boolean expression : 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60022″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-178-1.png?resize=257%2C147&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(178-1)” width=”257″ height=”147″>
Question 3: 
Express the OR operator in terms of AND and NOT operator. 
Аnswer: 
(A . B)’ = A¯¯¯¯<script type=”math/tex” id=”MathJax-Element-1″>\overline { A }</script> + B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-2″>\overline { B }</script> 
(A¯¯¯¯<script type=”math/tex” id=”MathJax-Element-3″>\overline { A }</script> + B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-4″>\overline { B }</script>)’ = A + B<div class=”google-auto-placed ap_container” style=”width: 100%; height: auto; clear: both; text-align: center;”><ins data-ad-format=”auto” class=”adsbygoogle adsbygoogle-noablate” data-ad-client=”ca-pub-7398766921532682″ data-adsbygoogle-status=”done” style=”display: block; margin: auto; background-color: transparent; height: 280px;” data-ad-status=”filled”><div id=”aswift_4_host” style=”border: none; height: 280px; width: 750px; margin: 0px; padding: 0px; position: relative; visibility: visible; background-color: transparent; display: inline-block; overflow: visible;” tabindex=”0″ title=”Advertisement” aria-label=”Advertisement”><iframe id=”aswift_4″ name=”aswift_4″ style=”left:0;position:absolute;top:0;border:0;width:750px;height:280px;” sandbox=”allow-forms allow-popups allow-popups-to-escape-sandbox allow-same-origin allow-scripts allow-top-navigation-by-user-activation” width=”750″ height=”280″ frameborder=”0″ marginwidth=”0″ marginheight=”0″ vspace=”0″ hspace=”0″ allowtransparency=”true” scrolling=”no” src=”https://googleads.g.doubleclick.net/pagead/ads?client=ca-pub-7398766921532682&output=html&h=280&adk=2523109437&adf=2199795995&pi=t.aa~a.3071249423~i.15~rp.4&w=750&fwrn=4&fwrnh=100&lmt=1670938749&num_ads=1&rafmt=1&armr=3&sem=mc&pwprc=6908628465&ad_type=text_image&format=750×280&url=https%3A%2F%2Fwww.cbsetuts.com%2Fncert-solutions-class-12-computer-science-c-boolean-algebra%2F&fwr=0&pra=3&rh=188&rw=750&rpe=1&resp_fmts=3&wgl=1&fa=27&adsid=ChEIgPvEnQYQ7oKUkIGo7bvHARI9AMmKl4lSlmxSp_EuqkkvujLyJB4s6dU2LtSRXWDmFUOke_SQLWbIo9A98U6dlFYFIStmRtcRp8siPDvC_w&uach=WyJXaW5kb3dzIiwiMTAuMC4wIiwieDg2IiwiIiwiMTA4LjAuNTM1OS4xMjUiLFtdLGZhbHNlLG51bGwsIjY0IixbWyJOb3Q_QV9CcmFuZCIsIjguMC4wLjAiXSxbIkNocm9taXVtIiwiMTA4LjAuNTM1OS4xMjUiXSxbIkdvb2dsZSBDaHJvbWUiLCIxMDguMC41MzU5LjEyNSJdXSxmYWxzZV0.&dt=1672590903988&bpp=4&bdt=2098&idt=5&shv=r20221207&mjsv=m202212010101&ptt=9&saldr=aa&abxe=1&cookie=ID%3D70877dca39d7cee1-22c4db3411d900f8%3AT%3D1672348851%3ART%3D1672348851%3AS%3DALNI_MYJDyG_bu32YHEUMIGJIo2bVt3_WQ&gpic=UID%3D00000b9a5c88ec58%3AT%3D1672348851%3ART%3D1672575613%3AS%3DALNI_Mb87E1-6Z_pG79EP2w2fRKpjdy0KA&prev_fmts=0x0%2C300x600%2C750x280&nras=3&correlator=6454028235071&frm=20&pv=1&ga_vid=540223204.1672348852&ga_sid=1672590903&ga_hid=613024059&ga_fc=1&u_tz=330&u_his=5&u_h=768&u_w=1366&u_ah=728&u_aw=1366&u_cd=24&u_sd=1&dmc=4&adx=105&ady=1522&biw=1349&bih=657&scr_x=0&scr_y=0&eid=44759875%2C44759926%2C44759842%2C31071167%2C44779793%2C44780792%2C21065724&oid=2&pvsid=972740684238546&tmod=365514175&uas=3&nvt=1&ref=https%3A%2F%2Fwww.learncbse.in%2F&eae=0&fc=1408&brdim=0%2C0%2C0%2C0%2C1366%2C0%2C1366%2C728%2C1366%2C657&vis=1&rsz=%7C%7Cs%7C&abl=NS&fu=128&bc=31&jar=2022-12-31-10&ifi=5&uci=a!5&btvi=3&fsb=1&xpc=rAqdfhJm9e&p=https%3A//www.cbsetuts.com&dtd=154″ data-google-container-id=”a!5″ data-google-query-id=”CLiyl9XmpvwCFXiC6QUdnlMC-g” data-load-complete=”true”></iframe></div></ins></div>
Question 4: 
Specify which axioms/theorems are being used in the following Boolean reductions : 
(a) (be)’ + be = 1 
(b) xyz + zx = xz 
Аnswer: 
(a) x + x’ = 1 & Complementary law 
(b) y + x = x & Absorption law.
Question 5: 
State and verify Associative law using Truth Table. 
Аnswer: 
Associative Law: This law states that: 
(A + B) + C = A + (B + C) 
(A.B).C = A. (B.C) 
Proof: 
<img loading=”lazy” class=”alignnone size-full wp-image-60023″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-178-2.png?resize=864%2C228&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(178-2)” width=”750″ height=”198″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-178-2.png?w=864&ssl=1 864w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-178-2.png?resize=300%2C79&ssl=1 300w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-178-2.png?resize=768%2C203&ssl=1 768w” sizes=”(max-width: 864px) 100vw, 864px”> 
∴ From above truth table, 
(A + B) + C = A + (B + C) 
Similarly, we can prove, 
A. (B.C) = (A. B).C 
<img loading=”lazy” class=”alignnone size-full wp-image-60024″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-1.png?resize=806%2C232&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(179-1)” width=”806″ height=”232″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-1.png?w=806&ssl=1 806w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-1.png?resize=300%2C86&ssl=1 300w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-1.png?resize=768%2C221&ssl=1 768w” sizes=”(max-width: 806px) 100vw, 806px” data-recalc-dims=”1″>
Short Answer Type Questions-I    [2 mark each]
Question 1: 
Correct the following boolean statements : 
1. X+1 = X 
2. (A’)’ = A’ 
3.  A+A’ = 0 
4.  (A+B)’ = A.B 
Аnswer: 
1. X+l=l or X+0=X 
2. ((A’)’) = A 
3. A + A’ = 1 or A. A’ = 0 
4. (A 4- B)’ = A’.B1
Question 2: 
Write the POS form of a Boolean Function F, which is represented in a truth table as follows : 
<img loading=”lazy” class=”alignnone size-full wp-image-60026″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-2.png?resize=367%2C260&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(179-2)” width=”367″ height=”260″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-2.png?w=367&ssl=1 367w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-179-2.png?resize=300%2C213&ssl=1 300w” sizes=”(max-width: 367px) 100vw, 367px” data-recalc-dims=”1″> 
Аnswer: 
(P+Q+R).(P’+Q+R).(P’+Q’+R)
Short Answer Type Questions-II  [3 mark each]
Laws and Theorems 
Question 1: 
State and Verify Absorption law algebraically. 
Аnswer: 
Absorption law states that : 
A + AB = A and A. (A + B) = A 
Algebraic method : 
Taking LHS 
A + AB = (A.l) + (A.B) by Identity 
= A. (1 + B) by Distribution 
= A.l by Null Element 
= A
Question 2: 
State and define principle of duality. Why is it so important in Boolean Algebra ? 
Аnswer: 
Principle of duality : Duality principle states that from every boolean relation another boolean relation can be derived by : 
(i) Changing each OR sign (+) to an AND sign (-). 
(ii) Changing each AND sign (-) to an OR sign (+) 
ex : Dual of A + A’B = A. (A’ + B) 
Importance in Boolean Algebra : The principle of duality is an important concept in Boolean algebra, particularly in proving various theorems. The principle of duality is used extensively in proving Boolean algebra theorem. Once we prove that an expression is valid, by the principle of duality, its dual is also valid. Hence, our effort in proving various theorems is reduced to half.<div class=”google-auto-placed ap_container” style=”width: 100%; height: auto; clear: both; text-align: center;”><ins data-ad-format=”auto” class=”adsbygoogle adsbygoogle-noablate” data-ad-client=”ca-pub-7398766921532682″ data-adsbygoogle-status=”done” style=”display: block; margin: auto; background-color: transparent; height: 280px;”><div id=”aswift_5_host” style=”border: none; height: 280px; width: 750px; margin: 0px; padding: 0px; position: relative; visibility: visible; background-color: transparent; display: inline-block;”></div></ins></div>
Question 3: 
Name the law shown below & verify it using a . truth table. 
X+ X¯¯¯¯<script type=”math/tex” id=”MathJax-Element-5″>\overline { X }</script>.Y = X + Y. 
Аnswer: 
This law is called “Absorption Law” also referred as redundance law. 
<img loading=”lazy” class=”alignnone size-full wp-image-60027″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-1.png?resize=356%2C158&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-1)” width=”356″ height=”158″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-1.png?w=356&ssl=1 356w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-1.png?resize=300%2C133&ssl=1 300w” sizes=”(max-width: 356px) 100vw, 356px” data-recalc-dims=”1″><div class=”google-auto-placed ap_container” style=”width: 100%; height: auto; clear: both; text-align: center;”><ins data-ad-format=”auto” class=”adsbygoogle adsbygoogle-noablate” data-ad-client=”ca-pub-7398766921532682″ data-adsbygoogle-status=”done” style=”display: block; margin: auto; background-color: transparent; height: 280px;”><div id=”aswift_6_host” style=”border: none; height: 280px; width: 750px; margin: 0px; padding: 0px; position: relative; visibility: visible; background-color: transparent; display: inline-block;”></div></ins></div>
Question 4: 
Draw a logic circuit for the following Boolean expression : ab + c.d’. 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60028″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-2.png?resize=402%2C174&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-2)” width=”402″ height=”174″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-2.png?w=402&ssl=1 402w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-2.png?resize=300%2C130&ssl=1 300w” sizes=”(max-width: 402px) 100vw, 402px” data-recalc-dims=”1″>
Question 5: 
Write the SOP form of a Boolean function F, which is represented in a truth table as follows : 
<img loading=”lazy” class=”alignnone size-full wp-image-60029″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-3.png?resize=112%2C279&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-3)” width=”112″ height=”279″ data-recalc-dims=”1″> 
Аnswer: 
A’B’C + A’BC + AB’C + AB’C<div class=”google-auto-placed ap_container” style=”width: 100%; height: auto; clear: both; text-align: center;”><ins data-ad-format=”auto” class=”adsbygoogle adsbygoogle-noablate” data-ad-client=”ca-pub-7398766921532682″ data-adsbygoogle-status=”done” style=”display: block; margin: auto; background-color: transparent; height: 280px;”><div id=”aswift_7_host” style=”border: none; height: 280px; width: 750px; margin: 0px; padding: 0px; position: relative; visibility: visible; background-color: transparent; display: inline-block;”></div></ins></div>
Question 6: 
Draw the Logic Circuit for the following Boolean Expression : 
(U + V). w + z 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60030″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-4.png?resize=375%2C103&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-4)” width=”375″ height=”103″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-4.png?w=375&ssl=1 375w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-4.png?resize=300%2C82&ssl=1 300w” sizes=”(max-width: 375px) 100vw, 375px” data-recalc-dims=”1″>
Question 7: 
Verify the following using Boolean Laws : 
LT + V = LTV + LP.V + U.V 
Аnswer: 
L.H.S. 
= U’ + V 
= U’ . (V + V) + V (LP + U) 
= U’ . V + LP . V + U . V + U. V 
= U’. V + LP. V + U. V 
= R.H.S.
OR
R.H.S. 
= U’V’ + U’. V + U. V 
= LP . (V + V) + U. V 
= U’ 1 + U.V 
= U’ + U.V 
= U’ +V 
= L.H.S.
Question 8: 
Draw the Logic Circuit for the following Boolean Expression : 
(X’ + Y). Z + W’ 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60031″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-5.png?resize=405%2C114&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-5)” width=”405″ height=”114″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-5.png?w=405&ssl=1 405w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-5.png?resize=300%2C84&ssl=1 300w” sizes=”(max-width: 405px) 100vw, 405px” data-recalc-dims=”1″><div class=”google-auto-placed ap_container” style=”width: 100%; height: auto; clear: both; text-align: center;”><ins data-ad-format=”auto” class=”adsbygoogle adsbygoogle-noablate” data-ad-client=”ca-pub-7398766921532682″ data-adsbygoogle-status=”done” style=”display: block; margin: auto; background-color: transparent; height: 280px;”><div id=”aswift_8_host” style=”border: none; height: 280px; width: 750px; margin: 0px; padding: 0px; position: relative; visibility: visible; background-color: transparent; display: inline-block;”></div></ins></div>
Question 9: 
Write the equivalent boolean expression for the following logic circuit. 
<img loading=”lazy” class=”alignnone size-full wp-image-60032″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-6.png?resize=330%2C144&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-6)” width=”330″ height=”144″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-6.png?w=330&ssl=1 330w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-6.png?resize=300%2C131&ssl=1 300w” sizes=”(max-width: 330px) 100vw, 330px” data-recalc-dims=”1″> 
Аnswer: 
((X’.Y)’ + (X.Y’)’)’
Question 10: 
Write the equivalent Boolean Expression for the. following Logic Circuit : 
<img loading=”lazy” class=”alignnone size-full wp-image-60033″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-180-7.png?resize=251%2C99&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(180-7)” width=”251″ height=”99″ data-recalc-dims=”1″> 
Аnswer: 
Z = (A+B)(B’ +C) 
= A.B’ + AC + B.B’ + BC 
= A.B’ + AC + BC
Question 11: 
Obtain the Boolean Expression for the logic circuit shown below : 
<img loading=”lazy” class=”alignnone size-full wp-image-60034″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-1.png?resize=230%2C85&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(181-1)” width=”230″ height=”85″ data-recalc-dims=”1″> 
Аnswer: 
F = (A¯¯¯¯.B)+(C+D¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯)<script type=”math/tex” id=”MathJax-Element-6″>\left( \overline { A } .B \right) +\left( \overline { C+\overline { D } }\right)</script> 
= (A¯¯¯¯.B)+C¯¯¯¯.D<script type=”math/tex” id=”MathJax-Element-7″>\left( \overline { A } .B \right) +\overline { C } .D</script>
Question 12: 
Name the law shown below & verify it using a truth table. 
A + B . C = (A + B). (A + C). 
Аnswer: 
This law is called “Distributive Law”. 
Prove by Truth table 
<img loading=”lazy” class=”alignnone size-full wp-image-60035″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-2.png?resize=420%2C226&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(181-2)” width=”420″ height=”226″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-2.png?w=420&ssl=1 420w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-2.png?resize=300%2C161&ssl=1 300w” sizes=”(max-width: 420px) 100vw, 420px” data-recalc-dims=”1″>
Question 13: 
Obtain the Boolean Expression for the logic circuit shown below : 
<img loading=”lazy” class=”alignnone size-full wp-image-60036″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-3.png?resize=224%2C87&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(181-3)” width=”224″ height=”87″ data-recalc-dims=”1″> 
Аnswer: 
F = ( X.Y¯¯¯¯<script type=”math/tex” id=”MathJax-Element-8″>\overline { Y }</script> ) + ( Z¯¯¯¯<script type=”math/tex” id=”MathJax-Element-9″>\overline { Z }</script> + W). 
F = X¯¯¯¯<script type=”math/tex” id=”MathJax-Element-10″>\overline { X }</script> + Y + Z¯¯¯¯<script type=”math/tex” id=”MathJax-Element-11″>\overline { Z }</script> + W.
Question 14: 
State Demorgan’s law. Verify one of them using truth table. 
Аnswer: 
There are two Demorgan’s law : 
(i)  A.B¯¯¯¯¯¯¯¯¯¯<script type=”math/tex” id=”MathJax-Element-12″>\overline { A.B }</script> = A¯¯¯¯<script type=”math/tex” id=”MathJax-Element-13″>\overline { A }</script> + B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-14″>\overline { B }</script> 
(ii)  A+B¯¯¯¯¯¯¯¯¯¯¯¯¯<script type=”math/tex” id=”MathJax-Element-15″>\overline { A+B }</script> = XA¯¯¯¯¯¯¯¯<script type=”math/tex” id=”MathJax-Element-16″>\overline { XA}</script> . B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-17″>\overline { B }</script> 
Proof : 
(i)  A.B¯¯¯¯¯¯¯¯¯¯<script type=”math/tex” id=”MathJax-Element-18″>\overline { A.B }</script> = A¯¯¯¯<script type=”math/tex” id=”MathJax-Element-19″>\overline { A }</script> + B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-20″>\overline { B }</script> 
 
<img loading=”lazy” class=”alignnone size-full wp-image-60037″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-4.png?resize=367%2C136&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(181-4)” width=”367″ height=”136″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-4.png?w=367&ssl=1 367w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-4.png?resize=300%2C111&ssl=1 300w” sizes=”(max-width: 367px) 100vw, 367px” data-recalc-dims=”1″>
Question 15: 
Draw a logic Circuit for the boolean expression: 
A . B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-21″>\overline { B }</script> + (C + B¯¯¯¯<script type=”math/tex” id=”MathJax-Element-22″>\overline { B }</script>). A¯¯¯¯<script type=”math/tex” id=”MathJax-Element-23″>\overline { A }</script> 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60038″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-5.png?resize=347%2C186&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(181-5)” width=”347″ height=”186″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-5.png?w=347&ssl=1 347w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-5.png?resize=300%2C161&ssl=1 300w” sizes=”(max-width: 347px) 100vw, 347px” data-recalc-dims=”1″>
Question 16: 
Obtain the Boolean Expression for the logic circuit shown below : 
<img loading=”lazy” class=”alignnone size-full wp-image-60039″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-6.png?resize=329%2C241&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(181-6)” width=”329″ height=”241″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-6.png?w=329&ssl=1 329w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-181-6.png?resize=300%2C220&ssl=1 300w” sizes=”(max-width: 329px) 100vw, 329px” data-recalc-dims=”1″> 
F = P’ Q + (Q + R’) 
= Q. (P’ + R’)
Question 17: 
Verify the following using Boolean Laws X + Z = X + X’. Z + Y. Z 
Аnswer: 
Taking RHS 
X + X’Z + YZ 
= (X + X’). (X + Z) + YZ (Distribution Law) 
= 1. (X + Z) + YZ    (A + A’ = 1) 
= X + Z + YZ 
= X + Z (1 + Y) 
= X + Z    (1 + A = 1; 1. A = A) 
= Hence verified
Question 18: 
Verify the following using Boolean Laws : A + C = A + A. C + B.C 
Аnswer: 
A + C = A + A’.C + BC 
Solve RHS 
A + A’C + BC 
(A + A). (A + C) + BC [Using distributive law] 
1. (A + C) + BC 
= A + C + BC 
= A + C(1 + B) 
= A + C.1 
= A + C 
= LHS (Hence, verified)
Question 19: 
Obtain the Boolean Expression for the logic circuit shown below : 
<img loading=”lazy” class=”alignnone size-full wp-image-60040″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-1.png?resize=352%2C202&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-1)” width=”352″ height=”202″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-1.png?w=352&ssl=1 352w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-1.png?resize=300%2C172&ssl=1 300w” sizes=”(max-width: 352px) 100vw, 352px” data-recalc-dims=”1″> 
Expression at F : 
( X¯¯¯¯<script type=”math/tex” id=”MathJax-Element-24″>\overline { X }</script> . Y) + (Y + Z¯¯¯¯<script type=”math/tex” id=”MathJax-Element-25″>\overline { Z }</script> ) 
( X¯¯¯¯<script type=”math/tex” id=”MathJax-Element-26″>\overline { X }</script> + 1) Y + Z¯¯¯¯<script type=”math/tex” id=”MathJax-Element-27″>\overline { Z }</script>    [Distributive law] 
Y + Z¯¯¯¯<script type=”math/tex” id=”MathJax-Element-28″>\overline { Z }</script>    [ ∴ 1 + Z = 1]
Question 20: 
Verify the following using truth table : 
(i) X . X’ = 0 
(ii) X + 1 = 1 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60041″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-2.png?resize=272%2C204&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-2)” width=”272″ height=”204″ data-recalc-dims=”1″>
Question 21: 
Write the equivalent boolean expression for the following logic circuit : 
<img loading=”lazy” class=”alignnone size-full wp-image-60042″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-3.png?resize=266%2C94&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-3)” width=”266″ height=”94″ data-recalc-dims=”1″> 
Аnswer: 
Y = U V¯¯¯¯<script type=”math/tex” id=”MathJax-Element-29″>\overline { V }</script> + U¯¯¯¯<script type=”math/tex” id=”MathJax-Element-30″>\overline { U}</script> W¯¯¯¯¯<script type=”math/tex” id=”MathJax-Element-31″>\overline { W }</script>
Question 22: 
Write the equivalent boolean expression for the following logic circuit : 
<img loading=”lazy” class=”alignnone size-full wp-image-60043″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-4.png?resize=259%2C98&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-4)” width=”259″ height=”98″ data-recalc-dims=”1″> 
Аnswer: 
Y = (U + V¯¯¯¯<script type=”math/tex” id=”MathJax-Element-32″>\overline { V }</script> ). (U + W¯¯¯¯¯<script type=”math/tex” id=”MathJax-Element-33″>\overline { W }</script> )
Question 23: 
Verify the following using truth table : 
(i) X + 0 = X 
(ii) X + X’ = 1 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60044″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-5.png?resize=281%2C117&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-5)” width=”281″ height=”117″ data-recalc-dims=”1″> 
<img loading=”lazy” class=”alignnone size-full wp-image-60045″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-6.png?resize=272%2C131&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-6)” width=”272″ height=”131″ data-recalc-dims=”1″>
Question 24: 
Derive a Canonical SOP expression for a Boolean function F, represented by the following truth table : 
<img loading=”lazy” class=”alignnone size-full wp-image-60046″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-7.png?resize=259%2C288&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-7)” width=”259″ height=”288″ data-recalc-dims=”1″> 
Аnswer: 
F(A, B, C) = A’B’C + A’BC + AB’C + ABC 
OR 
F(A,B,C) =∑(0, 3,4,7)
Question 25: 
Derive a Canonical POS expression for a Boolean function F, represented by the following truth table : 
<img loading=”lazy” class=”alignnone size-full wp-image-60047″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-182-8.png?resize=300%2C266&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(182-8)” width=”300″ height=”266″ data-recalc-dims=”1″> 
Аnswer: 
F(RQ,R) = (P+Q+R’)(P+Q,+R)(P’+Q,+R’) (P’+Q’+R) 
OR 
F(RQ,R)=II(1,2,5,6)
Question 26: 
Obtain a simplified form for a Boolean expression : 
F(U, V, W, Z) = II (0,1,3,5, 6, 7,15) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60048″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-1.png?resize=417%2C247&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(183-1)” width=”417″ height=”247″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-1.png?w=417&ssl=1 417w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-1.png?resize=300%2C178&ssl=1 300w” sizes=”(max-width: 417px) 100vw, 417px” data-recalc-dims=”1″> 
(u + v + w).(u+z’).(v’+w’).(u’+w’+z)
Question 27: 
Reduce the following Boolean Expression to its simplest form using K-Map : 
F (X, X Z, W) = X (0,1, 6, 8, 9,10,11,12,15) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60049″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-2.png?resize=348%2C324&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(183-2)” width=”348″ height=”324″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-2.png?w=348&ssl=1 348w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-2.png?resize=300%2C279&ssl=1 300w” sizes=”(max-width: 348px) 100vw, 348px” data-recalc-dims=”1″> 
<img loading=”lazy” class=”alignnone size-full wp-image-60050″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-3.png?resize=351%2C342&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(183-3)” width=”351″ height=”342″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-3.png?w=351&ssl=1 351w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-3.png?resize=300%2C292&ssl=1 300w” sizes=”(max-width: 351px) 100vw, 351px” data-recalc-dims=”1″> 
Simplified Expression : XY + Y’Z’ + XZ’W’ + XZW + X’YZW’
Question 28: 
Reduce the following Boolean Expression to its simplest form using K-Map : 
F(X, Y, Z, W) = X (0,1,4, 5,6, 7,8, 9,11,15) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60051″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-4.png?resize=351%2C323&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(183-4)” width=”351″ height=”323″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-4.png?w=351&ssl=1 351w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-183-4.png?resize=300%2C276&ssl=1 300w” sizes=”(max-width: 351px) 100vw, 351px” data-recalc-dims=”1″> 
<img loading=”lazy” class=”alignnone size-full wp-image-60052″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-1.png?resize=350%2C341&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(184-1)” width=”350″ height=”341″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-1.png?w=350&ssl=1 350w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-1.png?resize=300%2C292&ssl=1 300w” sizes=”(max-width: 350px) 100vw, 350px” data-recalc-dims=”1″> 
Simplified Expression : Y’Z’ + XY + XZW
Question 29: 
Verify the following using Boolean Laws. 
X + Y’ = X. Y + X. Y + X’. Y 
Аnswer: 
L. H. S. 
= X + Y’ 
= X. (Y+Y’) + (X + X’). Y’ 
= X. Y + X. Y’ + X. Y’ + X’. Y’ 
= X. Y + X. Y’ + X’. Y’ 
= R. H. S 
OR 
= X. Y + X. Y’ + X’. Y’ 
= X. (Y + Y’) + X’. Y’ 
= X. 1 + X’. Y’ 
= X + X’. Y’ 
= X + Y 
= L. H. S
Question 30: 
State Distributive law and verify it using truth table. 
Аnswer: 
Distributive law : This law states that 
(i) x(y + z) = xy + x.z. 
(ii) x + yz = (x + y)(x + z) 
<img loading=”lazy” class=”alignnone size-full wp-image-60053″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-2.png?resize=426%2C605&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(184-2)” width=”426″ height=”605″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-2.png?w=426&ssl=1 426w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-2.png?resize=211%2C300&ssl=1 211w” sizes=”(max-width: 426px) 100vw, 426px” data-recalc-dims=”1″>
Question 31: 
Reduce the following Boolean Expression using KMap : 
F(A, B, C, D) = ∑{0,1,3, 5, 6, 7,9,11,13,14,15} 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60054″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-3.png?resize=368%2C361&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(184-3)” width=”368″ height=”361″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-3.png?w=368&ssl=1 368w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-184-3.png?resize=300%2C294&ssl=1 300w” sizes=”(max-width: 368px) 100vw, 368px” data-recalc-dims=”1″> 
A’B’C’+D+BC
<h3 style=”text-align: center;”>TOPIC-2 
 Karnaugh Map Minimization and Applications of Boolean Algebra</h3>
Very Short Answer Type Questions [1 mark each]
Question 1: 
Write Product of Sum expression of the function F (a, b, c, d) from the given truth table 
<img loading=”lazy” class=”alignnone size-full wp-image-60055″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-1.png?resize=365%2C388&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(186-1)” width=”365″ height=”388″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-1.png?w=365&ssl=1 365w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-1.png?resize=282%2C300&ssl=1 282w” sizes=”(max-width: 365px) 100vw, 365px” data-recalc-dims=”1″> 
Аnswer: 
F (a, b, c, d) = 
(a + b + c + d).(a + b + c + d’). (a + b’ + c + d) . (a + b’ + c’ + d’). (a’ + b + c + d). 
(a’ + b + c + d’). (a’ + b’ + c + d). (a’ + b’ + c + d’) . (a’ + b’ + c’ + d)
Question 2: 
Convert the following boolean expression inti! its equivalent Canonical Sum of Products form (SOP) : 
(U + V + W) (U + V + W’) (U’ + V + W) (U’ + V’ + W’) 
Аnswer: 
π (0,1, 4, 7) 
∑(2, 3, 5, 6) 
010 011 101 110 
= U’VW’ + U’VW + UV’W + UVW’
Question 3: 
Write the Sum of Product form of the function F(R Q, R) for the following truth table representation of F : 
<img loading=”lazy” class=”alignnone size-full wp-image-60056″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-2.png?resize=407%2C280&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(186-2)” width=”407″ height=”280″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-2.png?w=407&ssl=1 407w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-2.png?resize=300%2C206&ssl=1 300w” sizes=”(max-width: 407px) 100vw, 407px” data-recalc-dims=”1″>
Question 4: 
Write the Product of Sum form of the function F(X, Y, Z) for the following truth table representation of F : 
<img loading=”lazy” class=”alignnone size-full wp-image-60057″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-3.png?resize=402%2C282&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(186-3)” width=”402″ height=”282″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-3.png?w=402&ssl=1 402w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-3.png?resize=300%2C210&ssl=1 300w” sizes=”(max-width: 402px) 100vw, 402px” data-recalc-dims=”1″>
Question 5: 
Write the Product of Sum form of the function G(U, V W) for the following truth table representation of G : 
<img loading=”lazy” class=”alignnone size-full wp-image-60058″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-4.png?resize=409%2C313&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(186-4)” width=”409″ height=”313″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-4.png?w=409&ssl=1 409w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-4.png?resize=300%2C230&ssl=1 300w” sizes=”(max-width: 409px) 100vw, 409px” data-recalc-dims=”1″>
Question 6: 
Write the Product of Sum form of function G(U, V, W) for the following truth table representation of G : 
<img loading=”lazy” class=”alignnone size-full wp-image-60059″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-5.png?resize=369%2C274&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(186-5)” width=”369″ height=”274″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-5.png?w=369&ssl=1 369w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-186-5.png?resize=300%2C223&ssl=1 300w” sizes=”(max-width: 369px) 100vw, 369px” data-recalc-dims=”1″> 
<img loading=”lazy” class=”alignnone size-full wp-image-60060″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-1.png?resize=407%2C416&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(187-1)” width=”407″ height=”416″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-1.png?w=407&ssl=1 407w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-1.png?resize=294%2C300&ssl=1 294w” sizes=”(max-width: 407px) 100vw, 407px” data-recalc-dims=”1″>
Question 7: 
Write the Sum of Product form of the function F(A, B, C) for the following truth table reprsentation of 
<img loading=”lazy” class=”alignnone size-full wp-image-60061″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-2.png?resize=366%2C326&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(187-2)” width=”366″ height=”326″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-2.png?w=366&ssl=1 366w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-2.png?resize=300%2C267&ssl=1 300w” sizes=”(max-width: 366px) 100vw, 366px” data-recalc-dims=”1″> 
<img loading=”lazy” class=”alignnone size-full wp-image-60062″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-3.png?resize=404%2C218&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(187-3)” width=”404″ height=”218″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-3.png?w=404&ssl=1 404w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-3.png?resize=300%2C162&ssl=1 300w” sizes=”(max-width: 404px) 100vw, 404px” data-recalc-dims=”1″> 
SOP = A’BC’ + A’BC + AB’C’ + ABC
Question 8: 
Write the SOP form of a boolean function F, which is represented in a truth table as follows: 
<img loading=”lazy” class=”alignnone size-full wp-image-60063″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-4.png?resize=363%2C188&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(187-4)” width=”363″ height=”188″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-4.png?w=363&ssl=1 363w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-4.png?resize=300%2C155&ssl=1 300w” sizes=”(max-width: 363px) 100vw, 363px” data-recalc-dims=”1″> 
Аnswer: 
F(X, Y, Z) = X’.Y’. Z’ + X’. Y. Z’ + X. Y’. Z’+ X.Y.Z
Question 9: 
Write the POS form of boolean function G, which is represented in a truth table as follows : 
<img loading=”lazy” class=”alignnone size-full wp-image-60064″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-5.png?resize=362%2C189&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(187-5)” width=”362″ height=”189″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-5.png?w=362&ssl=1 362w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-5.png?resize=300%2C157&ssl=1 300w” sizes=”(max-width: 362px) 100vw, 362px” data-recalc-dims=”1″> 
Аnswer: 
G (A, B, C) = (A + B + C). (A + B’ + C’). (A’ + B + C). (A’ + B + C’)
Short Answer Type Questions-II
Question 1: 
Obtain the minimal SOP form for the following Boolean expression using K-Map. 
F(A,B,C,D) = ∑ (0,2,3,5,7,8,10,1143,15) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60065″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-187-6.png?resize=263%2C248&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(187-6)” width=”263″ height=”248″ data-recalc-dims=”1″> 
Quad 1 = m0 + m2 + m8 + m10 = B’D’ 
Quad 2 = m3 + m7 + m15 + m11 = CD 
Quad 3 = m5 + m7 + m15 + m13 = BD 
Minimal SOP = B’D’ + CD + BD
Question 2: 
Reduce the following Boolean expression using 
K-Map : 
F(A,B,C,D) = 7r (0,1,2,4,5,6,8,10) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60066″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-1.png?resize=246%2C224&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(188-1)” width=”246″ height=”224″ data-recalc-dims=”1″> 
F(A, B, C, D) = π(0,1,2,4, 5, 6,8,10) F = (A + C).(A + D).(B + D)
Question 3: 
Reduce the following using K-Map : 
F (A, B,C,D) = ∑(1,3,4,5,6,7,12,13) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60067″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-2.png?resize=259%2C247&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(188-2)” width=”259″ height=”247″ data-recalc-dims=”1″>
Question 4: 
Reduce the following boolean expression using 
K-map. 
F(EQ,R,S) = 2(0,2,4,5,6,7,8,10,13,15). 
Аnswer: 
P¯¯¯¯<script type=”math/tex” id=”MathJax-Element-34″>\overline { P }</script> Q + Q¯¯¯¯<script type=”math/tex” id=”MathJax-Element-35″>\overline { Q }</script>S¯¯¯<script type=”math/tex” id=”MathJax-Element-36″>\overline { S }</script> + QS. 
<img loading=”lazy” class=”alignnone size-full wp-image-60068″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-3.png?resize=237%2C234&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(188-3)” width=”237″ height=”234″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-3.png?w=237&ssl=1 237w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-3.png?resize=100%2C100&ssl=1 100w” sizes=”(max-width: 237px) 100vw, 237px” data-recalc-dims=”1″>
Question 5: 
Reduce the following Boolean Expression using K-Map : 
F(P, Q, R, S) = ∑(1,2, 3,4,5, 6, 7, 8,10) 
Аnswer: 
<img loading=”lazy” class=”alignnone size-full wp-image-60069″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-4.png?resize=293%2C202&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(188-4)” width=”293″ height=”202″ data-recalc-dims=”1″> 
F(P, Q, R, S) = P’Q + P’S + P’R’S’ + PQ’S’
Question 6: 
Reduce the following Boolean Expression using 
K-Map : 
F (A, B, C, D) = ∑(2, 3,4,5, 6, 7,8,10,11) 
<img loading=”lazy” class=”alignnone size-full wp-image-60070″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-5.png?resize=226%2C432&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(188-5)” width=”226″ height=”432″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-5.png?w=226&ssl=1 226w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-188-5.png?resize=157%2C300&ssl=1 157w” sizes=”(max-width: 226px) 100vw, 226px” data-recalc-dims=”1″> 
F (A, B, C, D) = A’B + A’C + B’C + ABD’
Long Answer Type Questions [4 marks each]
Question 1: 
Verify the following using Boolean Laws : 
[Delhi, 2016] 
A ‘ + B’ . C=A’ . B ‘ . C ‘ + A’ . B . C ‘ + A’ .B.C + A’ .B’ .C+ A.B’ .C 
Аnswer: 
A’+ B’C = A’B’C’ + A,BCI + A’BC’ + A’BC + A’B’C + ABC 
=A’C'(B’+B)+A’C (Grouping) 
(B+B’)+AB’C 
=> A’C’+ A’C + AB’C 
(x+x’y=x+y) 
=> A’ (C+C’ ) +AB’ C 
=> A’+AB’C 
(x+x’=1) 
=> A’+B’C 
X=A’ y=B’C 
= LHS 
Hence Proved.
Question 2: 
Write the Boolean Expression for the result of the Logic Circuit as shown below : 
<img loading=”lazy” class=”alignnone size-full wp-image-60071″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-1.png?resize=419%2C207&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(189-1)” width=”419″ height=”207″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-1.png?w=419&ssl=1 419w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-1.png?resize=300%2C148&ssl=1 300w” sizes=”(max-width: 419px) 100vw, 419px” data-recalc-dims=”1″> 
Аnswer: 
F = (u+v’).(u+w).(v+w’)
Question 3: 
Derive a Canonical POS expression for a Boolen function F, represented by the following truth table : 
<img loading=”lazy” class=”alignnone size-full wp-image-60073″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-2.png?resize=397%2C217&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(189-2)” width=”397″ height=”217″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-2.png?w=397&ssl=1 397w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-2.png?resize=300%2C164&ssl=1 300w” sizes=”(max-width: 397px) 100vw, 397px” data-recalc-dims=”1″> 
Аnswer: 
F = ∑(0, 3,4,5) 
= (P + Q + R) (P + Q’ + R’) (P’ + Q + R) (P’ + Q + R’)
Question 4: 
Reduce the following Boolean Expression to its simplest form using K-Map : 
F (X, Y, Z, W) 
∑(2,6,7,8,9,10,11,13,14,15) 
Аnswer: 
∑(2,6,7,8,9,10,11,13,14,15) 
<img loading=”lazy” class=”alignnone size-full wp-image-60074″ src=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-3.png?resize=316%2C212&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(189-3)” width=”316″ height=”212″ srcset=”https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-3.png?w=316&ssl=1 316w, https://i2.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-3.png?resize=300%2C201&ssl=1 300w” sizes=”(max-width: 316px) 100vw, 316px” data-recalc-dims=”1″> 
F = XY’ + ZW’+ XW + YZ
Question 5: 
Write the Boolean Expression for the result of the Logic Circuit as shown below : 
<img loading=”lazy” class=”alignnone size-full wp-image-60075″ src=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-4.png?resize=425%2C199&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(189-4)” width=”425″ height=”199″ srcset=”https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-4.png?w=425&ssl=1 425w, https://i1.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-4.png?resize=300%2C140&ssl=1 300w” sizes=”(max-width: 425px) 100vw, 425px” data-recalc-dims=”1″> 
Аnswer: 
G = PQ’ + PR + QR’
Question 6: 
Derive a Canonical SOP expression for a Boolean function G, represented by the following truth table : 
<img loading=”lazy” class=”alignnone size-full wp-image-60076″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-5.png?resize=385%2C251&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(189-5)” width=”385″ height=”251″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-5.png?w=385&ssl=1 385w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-189-5.png?resize=300%2C196&ssl=1 300w” sizes=”(max-width: 385px) 100vw, 385px” data-recalc-dims=”1″> 
Аnswer: 
G = ∑{(0,2,6,7) 
= A’B’C’ + A’BC’ + ABC’ + ABC
Question 7: 
Verify the following using Boolean Laws : 
X’+ Y’Z = X’ .Y’ .Z’+X’ .Y.Z’+X’ .Y.Z+X’ .Y’ .Z + X.Y’.Z. 
Аnswer: 
X ‘+Y’ Z=X ‘ Y ‘ Z ‘ +X ‘ YZ ‘ +X ‘ YZ+X ‘ Y ‘ Z+XY ‘ Z 
Taking RHS 
Grouping terms 
=> x’Z’ (Y’+Y)+X’ Z(Y+Y’)+XY’Z 
=> X’Z’+X’Z+XY’Z 
(Y+Y’=1) 
=> X'(Z’+Z)+XY’Z 
(Grouping) 
=> X’+XY’Z (Z+Z’=l) 
=> X’+Y’Z (Substitute X=X’ Y=Y’Z X+X’ Y = X+Y) 
= LHS
Question 8: 
Reduce the following Boolean Expression to its simplest form using K-Map : 
F(P,Q,R,S) = ∑(0,4,5,8,9,10,11,12,13,15) 
Аnswer: 
F(P,Q,R,S) = ∑(0,4,5,8,9,10,11,12,13,15) 
<img loading=”lazy” class=”alignnone size-full wp-image-60077″ src=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-190-1.png?resize=391%2C281&ssl=1″ alt=”ncert-solutions-class-12-computer-science-c-boolean-algebra-(190-1)” width=”391″ height=”281″ srcset=”https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-190-1.png?w=391&ssl=1 391w, https://i0.wp.com/www.cbsetuts.com/wp-content/uploads/2017/12/ncert-solutions-class-12-computer-science-c-boolean-algebra-190-1.png?resize=300%2C216&ssl=1 300w” sizes=”(max-width: 391px) 100vw, 391px” data-recalc-dims=”1″> 
F = R’ S ‘ + PQ ‘ + QR’ + PS
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NCERT Solutions for Class 12 Computer Science (C++) – Boolean Algebra

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