ITPEC FE Morning April 2021 Question 1
Which of the following is equivalent to the logic expression below?
(x + y) · (x + z)
Here, the letters are logic variables; x + y, x · y, and x are OR, AND, and NOT operations on the corresponding variables, respectively.
Explanation
x + y · z — OR distributes over AND, so the common term x factors out of both brackets.
Expand the product with the distributive law: (x + y) · (x + z) = x · x + x · z + y · x + y · z.
Idempotence gives x · x = x, and absorption removes every term that contains x alongside a lone x: x + x · z + x · y = x. What remains is x + y · z.
Check with values: if x = 1 both sides are 1; if x = 0 the left side is y · z and so is the right.
Answer (b)
Why not others:
- •(a) — x · (y + z) is the other distributive law, AND over OR; with x = 1, y = z = 0 it gives 0 while the original gives 1
- •(c) — x · y + y · z also gives 0 for x = 1, y = z = 0: it loses the lone x
- •(d) — (x + y) · z is 0 whenever z = 0, yet the original is 1 for x = 1, z = 0
Key rule: In Boolean algebra OR distributes over AND as well as AND over OR: (x + y) · (x + z) = x + y · z; test a candidate with x = 1 and the rest 0 to rule options out fast.
Basic Theory · AI-written, may contain errors
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