Consider the Boolean operator # with the following properties: x # 0 = x, x #…
2016
Consider the Boolean operator # with the following properties:
x # 0 = x, x # 1 = x̄, x # x = 0 and x # x̄ = 1. Then x # y is equivalent to
- A.
xȳ + x̄y
- B.
xȳ + x̄ȳ
- C.
x̄y + xy
- D.
xy + x̄ȳ
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Correct answer: A
Final expression: x # y = x ȳ + x̄ y (exclusive OR).
Reasoning:
From x # 0 = x: when y = 0 the operator must return x.
From x # 1 = x̄: when y = 1 the operator must return the complement of x.
From x # x̄ = 1: when y = x̄ the operator must return 1.
The function that meets these conditions is the exclusive OR. Verify by substituting into x ȳ + x̄ y:
If y = 0: x ȳ + x̄ y = x·1 + x̄·0 = x.
If y = 1: x ȳ + x̄ y = x·0 + x̄·1 = x̄.
If y = x̄: both terms become 1 for the matching case, so x ȳ + x̄ y = 1.
Therefore the operator is x ȳ + x̄ y, which is the exclusive OR of x and y.
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