GATE ECE 2022 | Question: 17

Question

Select the Boolean function(s) equivalent to $x+yz,$ where $x, y,$ and $z$ are Boolean variables, and $+$ denotes logical $\text{OR}$ operation.

• A. $x + z + xy$
• B. $(x + y)(x + z)$
• C. $x + xy + yz$
• D. $x + xz + xy$

Answer 1

Detailed Video Solution: https://youtu.be/OMayy6p6Hfs 

Given boolean function is $f = x + yz$

Option A:

$x + z + xy = x + xy + z = x + z$ ; (We used Absorption Property which is $A + AB = A$)

So, Option A is Not equivalent to $f.$

Option B:

$x + yz = (x+y)(x+z) $ ;  (It is the Distributive Property which is $A + BC = (A+B)(A+C)$)

So, Option B is equivalent to $f.$

Option C:

$x + xy + yz = x + yz$  ; (We used Absorption Property which is $A + AB = A$)

So, Option C is equivalent to $f.$

Option D:

$x + xy + xz = x$  ; (We used Absorption Property which is $A + AB = A$)

So, Option D is Not equivalent to $f.$

Final answer is Option B,C.