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  1. Let $*$ be a Boolean operation defined as $A*B = AB + \overline{A}\;\overline{B}$. If $C=A*B$ then evaluate and fill in the blanks:
    1. $A*A=$____
    2. $C*A=$____
  2. Solve the following boolean equations for the values of $A, B$ and $C$:
    $AB+\overline{A}C=1$
    $AC+B=0$

asked in Digital Logic by Veteran (59.8k points)
edited by | 606 views
+1

Key Point: * is ex-nor Boolean operator, and it is commutative as well as associative.

1 Answer

+18 votes
Best answer
a) i) $A$*$ A = AA+A'A' = A + A' = 1$

    ii) $C$*$A= (A*B)*A = (AB + A'B')$*$A= (AB + A'B')A +(AB + A'B')'A'$

            $=(AB + A'B')A+(A'B+AB')A' = AB +0 +A'B+0 = B.$

b) $AB + A'C= 1$, $AC + B = 0$

$AC + B = 0$, means both $B = 0$  and $AC = 0$

$AB+ A'C = 1$

$A'C = 1$       [ bcoz $B = 0$ so, $AB = 0$ ]

So, $C = 1$ and $A = 0$

So, $A = 0$ , $B = 0$ and $C = 1$
answered by Veteran (55.8k points)
edited by
0

Sir , plz explain this part  once more 

AB + A'C= 1 , AC + B = 0

+5
AC+ B , means OR, OR is 0 only when both inputs in OR are 0

means B is 0 and AC is 0

AB+AC'= 1

as B is 0 already known now, then AB is also 0.

OR is 1 , when atleast one of input is 1, but AB is 0 so A'C should be 1

now A'C =1 , this is AND of A' and C. AND is 1 , when both inputs are 1

mean A' =1 and C =1

A'=1 , means A = 0
+1

PART a. :

here boolean operation * denotes the equivalence function $\color{RED}{⊙}$.
It evaluates to true only when both the variables have same values.
$F(x,y)=x\odot y= \left\{\begin{matrix} 1 & x=y=1 \ or \ x=y=0 \\ 0&Otherwise \end{matrix}\right.$

So,
$i) A* A  \rightarrow always \ 1$ 
$A=0 , \ \  \ \  0*0=1$ 
$A=1 , \ \ \ \  1*1=1$

$ii) C∗A$
$A∗B*A = A*A*B = 1*B = B$       (equivalence operation is both commutative and associative) 
$B=0 , \ \  \ \  1*0=0$ 
$B=1 , \ \ \ \  1*1=1$

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