ITPEC FE Morning April 2018 Question 2
ITPEC FE Morning April 2018 — Question 2 of 80
Distributive Law of Sets — tests whether you can recognize the set-theoretic distributive law.
The distributive laws for sets state:
- •Intersection distributes over union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- •Union distributes over intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Option (d) states:
(A ∩ C) ∪ (B ∩ C) = (A ∪ B) ∩ C
Factor out C from the left side using the first distributive law (with C as the common factor):
(A ∩ C) ∪ (B ∩ C) = (A ∪ B) ∩ C ✓
This is a direct application of the distributive property with C distributed across the union (A ∪ B).
Why not others:
- (a) (A ∪ B) ∩ (A ∩ C) simplifies to A ∩ C by absorption, while B ∩ (A ∪ C) does not equal A ∩ C in general.
- (b) (A ∪ B) ∩ C ≠ (A ∪ C) ∩ (B ∪ C); the right side expands to (A ∩ B) ∪ C by the second distributive law.
- (c) Left side simplifies to A ∩ (B ∪ C), right side to B ∩ (A ∪ C); these are not equal in general.
Key rule: Intersection distributes over union and vice versa — factor out the common set just like factoring in algebra.
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