Question

In boolean algebra, the unit element $1$ is-

• A. has two value
• B. is unique
• C. has at least two values
• D. None of these

Answer 1

The correct option is B

is unique

Explanation for the correct option:

Proving that the unit element in Boolean algebra is unique:

Two binary operations are defined on Boolean algebra, plus, and multiplication.

Let $a$ be an element of Boolean algebra, then,

$a\text{'}+a=0$ and $a\text{'}\text{'}·a=1$

where $a\text{'}$and $a\text{'}\text{'}$ are the compliments of $a$ with respect to plus and multiplication respectively.

Then, $0$ is called a zero element and $1$ is called a unit element in Boolean algebra.

and

$a+0=a=0+a\&a·1=a=1·a$

Let,$x_1$ and $x_2$ are two unit elements of Boolean algebra, and let $a$ be any element of Boolean algebra. Then,

$a·x_1=a=x_1·a\&a·x_2=a=x_2·a$

Since

$$\begin{gathered} a·x_1=a=x_2·a \\ \Rightarrow a·x_1=x_2·a \\ \Rightarrow x_1=x_2 \end{gathered}$$

Thus, in Boolean algebra, the unit element is unique.

Therefore, the correct answer is option (B).