UGC NET Computer Science Boolean Algebra Previous Year Questions (PYQs) – Page 2 of 2

Given Boolean expression is

$\overline{A}\cdot B+A\cdot \overline{B}+A\cdot B$

Now, group the last two terms:

$= \overline{A}\cdot B+A(\overline{B}+B)$

Since,

$\overline{B}+B=1$

So,

$= \overline{A}\cdot B+A$

Using Boolean identity,

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

Therefore,

$\overline{A}\cdot B+A\cdot \overline{B}+A\cdot B=A+B$

For equation $(i)$,

$wx+w(\overline{x}+y)+x(\overline{x}+y)$

$=wx+w\overline{x}+wy+x\overline{x}+xy$

Since,

$x\overline{x}=0$

So,

$=wx+w\overline{x}+wy+xy$

$=w(x+\overline{x})+wy+xy$

$=w+wy+xy$

$=w+xy$

But the right side is

$x+wy$

Since $w+xy$ is not always equal to $x+wy$, equation $(i)$ is false.

Now, for equation $(ii)$,

$(w\overline{x}(y+x\overline{z})+\overline{w}\overline{x})y$

$=(w\overline{x}y+w\overline{x}x\overline{z}+\overline{w}\overline{x})y$

Since,

$\overline{x}x=0$

So,

$=(w\overline{x}y+\overline{w}\overline{x})y$

$=w\overline{x}y+\overline{w}\overline{x}y$

$=\overline{x}y(w+\overline{w})$

$=\overline{x}y$

So, equation $(ii)$ is true.

Therefore, $(i)$ is false and $(ii)$ is true.

Statement A:

A finite Boolean algebra always has $2^n$ elements.

Possible number of elements can be:

$1,2,4,8,16,\ldots$

But $5$ is not of the form $2^n$.

So, statement A is false.

Statement B:

In Boolean algebra, every element has a unique complement.

So, statement B is true.

Statement C:

A Boolean algebra is complemented and distributive.

Also, every Boolean algebra is relatively complemented.

But statement C says it is not relatively complemented.

So, statement C is false.

Statement D:

The direct product of two Boolean algebras is also a Boolean algebra.

So, statement D is true.

Therefore, only B and D are correct.