The GATE Grind

GATE 2022 EC – Question 27

Digital Circuits · Combinatorial Circuits · 1 mark · Multiple select

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

  1. $x+z+xy$
  2. $(x+y)(x+z)$
  3. $x+xy+yz$
  4. $x+xz+xy$

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Show answer and explanation

Correct answer: (B) $(x+y)(x+z)$; (C) $x+xy+yz$

Explanation

By the distributive law, $(x+y)(x+z)=x+yz$, so B is equivalent. In C, $x+xy=x$, leaving $x+yz$ ✓. In A the extra term $z$ makes the function 1 when $x=0,y=0,z=1$, whereas $x+yz=0$ there. In D, $x+xz+xy=x$, which differs from $x+yz$ when $x=0,y=z=1$.