Answer
Proof by contradiction:
Let us suppose (A − C) ∩ (B − C) ∩ (A − B) ≠ ∅.
That is, suppose there exist sets A, B and C such that (A − C) ∩ (B − C) ∩ (A − B) ≠ ∅.
Then there is an element x in (A − C) ∩ (B − C) ∩ (A − B) such that x ≠ ∅.
By definition of intersection,
we should have,
x ∈ (A − C) and x ∈ (B − C) and x ∈ (A − B).
Applying the definition of intersection again,
we have that
Since x ∈ (A - C), x ∈ A and x ∉ C -(i)
Since x ∈ (B - C), x ∈ B and x ∉ C -(ii)
Since x ∈ (A - B), x ∈ A and x ∉ B -(iii)
From (ii) and (iii) we see that x ∈ B and x ∉ B which is a contradiction.
So our assumption is false,
Therefore (A − C) ∩ (B − C) ∩ (A − B) = ∅.
Work Step by Step
Steps:
1. Start with an assumption saying the statement is false.
that is (A − C) ∩ (B − C) ∩ (A − B) ≠ ∅
2. Breakdown the statements into sub-statements using definition of set intersection.
x ∈ (A - C) implies x ∈ A and x ∉ C and so on.
3. Show that the assumption is invalid with the sub-statements.