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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 $C$:
$AB+\overline{A}C=1$
$AC+B=0$

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+1

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

(A)

1. $A$*$A = AA+A'A' = A + A' = 1$
2. $C$*$A= (A*B)*A = (AB + A'B')$*$A= (AB + A'B')A +(AB + A'B')'A'$
$\qquad =(AB + A'B')A+(A'B+AB')A' = AB +0 +A'B+0 = B.$

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

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

$AB+ A'C = 1$

$A'C = 1$       $[\because B = 0, AB = 0]$

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

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

edited
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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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