Proof that (A ∩ C) − B ⊆ (A − B) ∩ (C − B) Consider the sentences in the following scrambled list. By definition of intersection x ∈ A ∩ C and x ∉ B. By definition of set difference x ∈ A ∩ C and x ∉ B. By definition of intersection, x ∈ (A − B) ∩ (C − B). By definition of set difference, x ∈ A and x ∈ C. So by definition of set difference, x ∈ A − B and x ∈ C − B. By definition of intersection, x ∈ A and x ∈ C. Hence both x ∈ A and x ∉ B and also x ∈ C, and x ∉ B. We prove part 2 by selecting appropriate sentences from the list and putting them in the correct order. Suppose x ∈ (A ∩ C) − B.

Question

Proof that (A ∩ C) − B ⊆ (A − B) ∩ (C − B)
Consider the sentences in the following scrambled list.
By definition of intersection x ∈ A ∩ C and x ∉ B.
By definition of set difference x ∈ A ∩ C and x ∉ B.
By definition of intersection, x ∈ (A − B) ∩ (C − B).
By definition of set difference, x ∈ A and x ∈ C.
So by definition of set difference, x ∈ A − B and x ∈ C − B.
By definition of intersection, x ∈ A and x ∈ C.
Hence both x ∈ A and x ∉ B and also x ∈ C, and x ∉ B.
We prove part 2 by selecting appropriate sentences from the list and putting them in the correct order.
Suppose x ∈ (A ∩ C) − B.

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