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What values of $A, B, C$ and $D$ satisfy the following simultaneous Boolean equations?

$\overline{A} + AB =0, AB=AC, AB+A\overline{C}+CD=\overline{C}D$

1. $A=1, B=0, C=0, D=1$

2. $A=1, B=1, C=0, D=0$

3. $A=1, B=0, C=1, D=1$

4. $A=1, B=0, C=0, D=0$

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0
AB+AC'+CD=C'D how we get this part can uh please explain !

$A'+AB=0 \implies A'+B=0$

$\therefore A'=0$ and $B=0$

$A=1$

$AB=AC \implies B=C\implies C=0$

$AB+AC'+CD=C'D$

$\implies 0+1+0=D$

$\implies D=1$
edited by
0

$AB=AC \rightarrow B=C$

Cancellation Law doesn't hold good for boolean algebra.
It holds only when -
$(AB=AC) \ \wedge (A=1) \rightarrow B=C$  \\doesn't hold when $A=0.$
and
$(A+B=A+C) \ \wedge (A=0) \rightarrow B=C$  \\doesn't hold when $A=1.$

For verification, just put up the values and check for AND, OR operations and their outputs.
+1 vote

"Options make this question pretty straight forward what we need to do just put the options and see which one satisfy the given condition

What if the same question is given in " Numeric" value where we have to find the value of Boolean variable A, B, C, D or asking to find the value in decimal where A(MsB) & D(Lsb) which make bit "tricky"

As, A' + AB = 0

A' can't be 1 because if A'=1 this implies 1+AB which is not 1 therefore, A'=0 => A=1

now AB has to be 0 to satisfy the equation, therefore B has to be 0 So , A=1, B=0

Now  AB=AC
as A=1, B=0 So C must be 0 then only AC will be 0. So, C=0

AB+AC'+CD=C'D

now AB=0 as B is 0 then AC'=1 (as A=1 , C'=1) , CD=0 as C is 0
So LHS is 1 now for RHS to be 1, C' and D both have to be 1 therefore D=1

So A=1, B=0, C=0, D=1
option a) is the ans.

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