The GATE Grind

GATE 2021 IN – Question 41

Digital Electronics · Number systems and combinational logic · 2 marks · Multiple choice

Boolean $F(X,Y,Z)=(X^\prime+Y+Z)(X+Y^\prime+Z^\prime)(X^\prime+Y+Z^\prime)(X^\prime Y^\prime Z^\prime+X^\prime YZ^\prime+XYZ^\prime)$. Which expression is true?

  1. $F=(X^\prime+Y+Z)(X^\prime+Y^\prime+Z^\prime)$
  2. $F=(X^\prime+Y)(X+Y^\prime+Z^\prime)$
  3. $F=X^\prime Z^\prime+YZ^\prime$
  4. $F=X^\prime Y^\prime Z+XYZ$

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Show answer and explanation

Correct answer: (C) $F=X^\prime Z^\prime+YZ^\prime$

Explanation

The last sum restricts possible true inputs to minterms 000,010,110, all with Z=0. Substitute each into the preceding three sum factors: each survives. Every other minterm is removed by the last factor.

Thus F=Σm(0,2,6)=Z′(X′+Y)=**X′Z′+YZ′, C**.