Question

Let $a$ be any element in Boolean algebra $B$, If $a+x=1$ and $ax=0$, then

• A. $x=1$
• B. $x=0$
• C. $x=a$
• D. $x=a\text{'}$

Answer 1

The correct option is D

$x=a\text{'}$

Explanation for correct answer:

Given, $a\in B=\{0,1\}$

since $B$ is a Boolean algebra

So, $a=0$or $a=1$

Substituting $a=0$ in the given equations we get

$$\begin{gathered} ax=0 \\ 0.x=0 \\ x=1 \end{gathered}$$

Also,

$$\begin{gathered} a+x=1 \\ 0+x=1 \\ x=1 \end{gathered}$$

Substituting $a=1$ in the given equations we get,

$$\begin{gathered} 1x=0 \\ 1.x=0 \\ x=0 \end{gathered}$$

Also,

$$\begin{gathered} 1+x=1 \\ x=1-1 \\ x=0 \end{gathered}$$

This shows that, $x=a\text{'}$ is true for the given two equations.

Hence, option (D) is the correct answer