Question

What will be the input of $A$ and $B$ respectively in order to satisfy the boolean expression, $\left(\overline{A+B}\right).\left(\overline{A.B}\right)=1?$

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

Answer 1

The correct option is A $0,0$

The given boolean expression can be written as,

$y=\left(\overline{A+B}\right).\left(\overline{A.B}\right)$

(Using the boolean algebra to convert product into addition and vice-versa)

$\Rightarrow y=(\overline{A}.\overline{B}).(\overline{A}+\overline{B})=(\overline{A}.\overline{A}).\overline{B}+\overline{A}.(\overline{B}.\overline{B})$

Using the boolean theorem, $A.A=A\text{and}A+A=A$
we get,

$y=\overline{A}.\overline{B}+\overline{A}.\overline{B}$

$\Rightarrow y=\overline{A}.\overline{B}$

According to question, $y=1$

$\Rightarrow \overline{A}.\overline{B}=1$

The truth table for $y=\overline{A}.\overline{B}$ is,

$\begin{array}{ccc}\text{A}& \text{B}& \text{Y}\\ 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\\ 1& 1& 0\end{array}$

$\therefore$ The output will $1$ when both the inputs are $0$.

Hence, option $\left(A\right)$ represents the correct input.

Why this question? Tip: In such problems, first simplify the expression using theorems of boolean algebra, thereafter construct the truth table for various combination of input values $(A,B)$.