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+2 votes
Simplify the following expression

AB’C + A’BC + A’B’C

Solution given is A’C + B’C can someone show me how?
in Digital Logic by Active (1.7k points) | 204 views
Draw a K map and simplify.

4 Answers

+3 votes

:AB'C + A'BC +A'B'C


=AB'C+A'C x 1


=C[(A+A')(B'+A')]                     [Distributive Law XY+Z==(X+Z)(Y+Z)]

=C x 1 x (B'+A')


by Loyal (5.2k points)
+1 vote

Let $f(A,B,C)=A\cdot\overline{B}\cdot C+\overline{A}\cdot B\cdot C+\overline{A}\cdot\overline{B}\cdot C$

Take common from first and last term

$f(A,B,C)=\overline{B}\cdot C\cdot(A+\overline{A})+\overline{A}\cdot B\cdot C$

$f(A,B,C)=\overline{B}\cdot C\cdot1+\overline{A}\cdot B\cdot C$                      ${\color{Red}{[X+\overline{X}=1]} }$

$f(A,B,C)=\overline{B}\cdot C+\overline{A}\cdot B\cdot C$

$f(A,B,C)=C(\overline{B}+\overline{A}\cdot B)$

$f(A,B,C)=C(\overline{B}+\overline{A})\cdot(\overline{B}+B)$                   ${\color{Magenta} {[X+Y\cdot Z=(X+Y)\cdot(X+Z)]}}$

$f(A,B,C)=C(\overline{A}+\overline{B})\cdot 1$

$f(A,B,C)=\overline{A}\cdot C+\overline{B}\cdot C$

by Veteran (58.7k points)
edited by
0 votes

$A\bar{B}C + \bar{A}BC + \bar{A}\bar{B}C$

Taking C common,

=$(A\bar{B} + \bar{A}B + \bar{A}\bar{B} ) . C$

=$((A \bigoplus B) + \bar{A}\bar{B} ) . C$

Remember this shortcut, [ $(P \bigoplus Q) + PQ  => P + Q $], also we know that $A \bigoplus B = \bar{A} \bigoplus \bar{B}$, negating both inputs have no effect on output of XOR and XNOR. Hence we can write

$(A \bigoplus B) + \bar{A}\bar{B} = \bar{A}+\bar{B} $


Hence it directly becomes,




Another method is to draw K-MAP from given min terms and then minimize :

$f(A,B,C) = A\bar{B}C + \bar{A}BC + \bar{A}\bar{B}C$

                               101          011           001

$f(A,B,C) = \sum \small m (1,3,5)$

by (457 points)
0 votes
AB’C + A’BC + A’B’C


(A+A')B'C+A'BC                   //A+A' = 1


(B'+A'B)C                      //(A+A'B) = A+B


by Junior (963 points)
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