The GATE Grind

GATE 2015 EC – Question 46

Digital Circuits · Combinatorial Circuits · 2 marks · Multiple choice

The Boolean expression $F(X, Y, Z) = \bar{X}Y\bar{Z} + X\bar{Y}\bar{Z} + XY\bar{Z} + XYZ$ converted into the canonical product of sum (POS) form is

  1. $(X + Y + Z)(X + Y + \bar{Z})(X + \bar{Y} + \bar{Z})(\bar{X} + Y + \bar{Z})$
  2. $(X + \bar{Y} + \bar{Z})(\bar{X} + Y + \bar{Z})(\bar{X} + \bar{Y} + Z)(\bar{X} + \bar{Y} + \bar{Z})$
  3. $(X + Y + Z)(\bar{X} + Y + \bar{Z})(\bar{X} + \bar{Y} + Z)(\bar{X} + \bar{Y} + \bar{Z})$
  4. $(X + \bar{Y} + \bar{Z})(\bar{X} + Y + Z)(\bar{X} + \bar{Y} + Z)(X + Y + Z)$

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Correct answer: (A) $(X + Y + Z)(X + Y + \bar{Z})(X + \bar{Y} + \bar{Z})(\bar{X} + Y + \bar{Z})$

Explanation

The function is 1 for the minterms 2, 4, 6 and 7. The product of sums lists the maxterms for the other four rows, which are 0, 1, 3 and 5. These are $(X + Y + Z)$, $(X + Y + \bar{Z})$, $(X + \bar{Y} + \bar{Z})$ and $(\bar{X} + Y + \bar{Z})$, which is option A.