Let A, B and C be non-empty sets and let X = (A - B) - C and Y = (A - C) - (B…
2005
Let A, B and C be non-empty sets and let X = (A - B) - C and Y = (A - C) - (B - C). Which one of the following is TRUE?
- A.
X = Y
- B.
X ⊂ Y
- C.
Y ⊂ X
- D.
none of these
Attempted by 430 students.
Show answer & explanation
Correct answer: A
Short answer: X = Y
Algebraic simplification:
X = (A − B) − C = A ∩ B^c ∩ C^c = A − (B ∪ C).
Y = (A − C) − (B − C) = A ∩ C^c ∩ (B ∩ C^c)^c = A ∩ C^c ∩ (B^c ∪ C) = A ∩ B^c ∩ C^c = A − (B ∪ C).
Since both X and Y equal A ∩ B^c ∩ C^c, they are equal.
Element-wise proof (optional):
If x ∈ X then x ∈ A, x ∉ B and x ∉ C. Hence x ∈ A − C and x ∉ (B − C), so x ∈ Y. Thus X ⊆ Y.
If x ∈ Y then x ∈ A and x ∉ C, and x ∉ (B − C). Because x ∉ C, the condition x ∉ (B − C) implies x ∉ B. Thus x ∈ A, x ∉ B and x ∉ C, so x ∈ X. Hence Y ⊆ X.
Combining both inclusions gives X = Y.