Answer
This statement is true.
Claim: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C).
Proof: Suppose A, B, and C are any sets. Let x ϵ A ⋂ (B – C). By definition of intersection, x ϵ A and x ϵ (B – C). By definition of set difference, x ϵ B and x C. Therefore, x ϵ A and x ϵ B, and x ϵ A and x C. Since x ϵ A and x ϵ B, then x ϵ (A ⋂ B) by definition of intersection. Since x ϵ A and x C, then x (A ⋂ C) by definition of intersection. By definition of set difference, x ϵ (A ∩ B) − (A ∩ C). Therefore, if x ϵ A ⋂ (B – C), then x ϵ (A ∩ B) − (A ∩ C). By definition of subset, A ⋂ (B – C) ⊆ (A ∩ B) − (A ∩ C).
Suppose A, B, and C are any sets. Let x ϵ (A ∩ B) − (A ∩ C). By definition of set differences, x ϵ (A ⋂ B) and x (A ⋂ C). By definition of intersection, x ϵ A and x ϵ B. By definition of intersection, x (A ⋂ C) implies that x C. Since x ϵ B and x C, then x ϵ (B – C) by definition of set difference. Since x ϵ A and x ϵ (B – C), x ϵ A ⋂ (B – C) by definition of intersection. Therefore, if x ϵ (A ∩ B) − (A ∩ C), then x ϵ A ⋂ (B – C). By definition of subset, (A ∩ B) − (A ∩ C) ⊆ A ⋂ (B – C).
Since A ⋂ (B – C) ⊆ (A ∩ B) − (A ∩ C) and (A ∩ B) − (A ∩ C) ⊆ A ⋂ (B – C), then A ∩ (B − C) = (A ∩ B) − (A ∩ C) by definition of subset equality.
Work Step by Step
Claim: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C).
Proof: Suppose A, B, and C are any sets. Let x ϵ A ⋂ (B – C). By definition of intersection, x ϵ A and x ϵ (B – C). By definition of set difference, x ϵ B and x C. Therefore, x ϵ A and x ϵ B, and x ϵ A and x C. Since x ϵ A and x ϵ B, then x ϵ (A ⋂ B) by definition of intersection. Since x ϵ A and x C, then x (A ⋂ C) by definition of intersection. By definition of set difference, x ϵ (A ∩ B) − (A ∩ C). Therefore, if x ϵ A ⋂ (B – C), then x ϵ (A ∩ B) − (A ∩ C). By definition of subset, A ⋂ (B – C) ⊆ (A ∩ B) − (A ∩ C).
Suppose A, B, and C are any sets. Let x ϵ (A ∩ B) − (A ∩ C). By definition of set differences, x ϵ (A ⋂ B) and x (A ⋂ C). By definition of intersection, x ϵ A and x ϵ B. By definition of intersection, x (A ⋂ C) implies that x C. Since x ϵ B and x C, then x ϵ (B – C) by definition of set difference. Since x ϵ A and x ϵ (B – C), x ϵ A ⋂ (B – C) by definition of intersection. Therefore, if x ϵ (A ∩ B) − (A ∩ C), then x ϵ A ⋂ (B – C). By definition of subset, (A ∩ B) − (A ∩ C) ⊆ A ⋂ (B – C).
Since A ⋂ (B – C) ⊆ (A ∩ B) − (A ∩ C) and (A ∩ B) − (A ∩ C) ⊆ A ⋂ (B – C), then A ∩ (B − C) = (A ∩ B) − (A ∩ C) by definition of subset equality.
