A Boolean function is x'y' + xy + x'y. Which of the following is equivalent? A. x' + y B. x + y' C. x + y D. x' + y'

GATE 2004 · Digital Logic · Boolean Algebra · easy

Answer: A. x' + y

1. Build truth table for f: (x=0,y=0): f = 1+0+0 = 1 (x=0,y=1): f = 0+0+1 = 1 (x=1,y=0): f = 0+0+0 = 0 (x=1,y=1): f = 0+1+0 = 1 So f is 0 only when x=1, y=0.
2. Algebraic simplification: Group x'y' and x'y: x'(y'+y) = x'*1 = x' Remaining: xy So f = x' + xy Absorption: x' + xy = x' + y (since x' + xy = x' + y by complement absorption) Verify: x'+y is 0 only when x=1,y=0. Matches truth table.
3. Verify answer A: x' + y: (x=0,y=0): 1+0=1; (x=0,y=1): 1+1=1; (x=1,y=0): 0+0=0; (x=1,y=1): 0+1=1. Matches f completely.