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

GATE 2005 · Discrete Mathematics · Set Theory · easy

Answer: X = Y — option A.

1. Expand membership condition for X: x ∈ X = (A-B)-C iff x ∈ (A-B) and x ∉ C iff (x ∈ A and x ∉ B) and x ∉ C iff x ∈ A and x ∉ B and x ∉ C.
2. Expand membership condition for Y and compare: x ∈ Y = (A-C)-(B-C) iff x ∈ (A-C) and x ∉ (B-C) iff (x ∈ A and x ∉ C) and not(x ∈ B and x ∉ C). The last part: not(x ∈ B and x ∉ C) = (x ∉ B) or (x ∈ C). Combined with x ∉ C: (x ∉ B or x ∈ C) and x ∉ C = x ∉ B and x ∉ C. So x ∈ Y iff x ∈ A and x ∉ B and x ∉ C. This is identical to x ∈ X. Hence X = Y.