Let 𝐴 and 𝐵 be sets. Show that:a) (𝐴 ∩ 𝐵) ⊆ 𝐴.b) 𝐴 ⊆ (𝐴 ∪ 𝐵).c) 𝐴 − 𝐵 ⊆ 𝐴.

Sure, here are the proofs for each of the statements:

a) (𝐴 ∩ 𝐵) ⊆ 𝐴. Proof: Let x be an element of (𝐴 ∩ 𝐵). By definition of intersection, this means that x is in both 𝐴 and 𝐵. Therefore, x is in 𝐴. So, we have shown that for any x in (𝐴 ∩ 𝐵), x is also in 𝐴. Hence, (𝐴 ∩ 𝐵) ⊆ 𝐴.

b) 𝐴 ⊆ (𝐴 ∪ 𝐵). Proof: Let x be an element of 𝐴. By definition of union, an element is in (𝐴 ∪ 𝐵) if it is in 𝐴 or it is in 𝐵 (or both). Since x is in 𝐴, it is in (𝐴 ∪ 𝐵). So, we have shown that for any x in 𝐴, x is also in (𝐴 ∪ 𝐵). Hence, 𝐴 ⊆ (𝐴 ∪ 𝐵).

c) 𝐴 − 𝐵 ⊆ 𝐴. Proof: Let x be an element of 𝐴 − 𝐵. By definition of set difference, this means that x is in 𝐴 and x is not in 𝐵. Therefore, x is in 𝐴. So, we have shown that for any x in 𝐴 − 𝐵, x is also in 𝐴. Hence, 𝐴 − 𝐵 ⊆ 𝐴.