Let 𝐴 and 𝐵 be sets. Show thata) (𝐴 ∩ 𝐵) ⊆ 𝐴.b) 𝐴 ⊆ (𝐴 ∪ 𝐵).c) 𝐴 − 𝐵 ⊆ 𝐴.d) 𝐴 ∩ (𝐵 − 𝐴) = ∅.e) 𝐴 ∪ (𝐵 − 𝐴) = 𝐴 ∪ 𝐵.

Sure, let's go through each of these step by step:

a) (𝐴 ∩ 𝐵) ⊆ 𝐴. This means that the intersection of sets A and B is a subset of A. By definition, the intersection of two sets is the set of elements that are common to both sets. Therefore, every element in (𝐴 ∩ 𝐵) is also in 𝐴. Hence, (𝐴 ∩ 𝐵) is a subset of 𝐴.

b) 𝐴 ⊆ (𝐴 ∪ 𝐵). This means that set A is a subset of the union of sets A and B. By definition, the union of two sets is the set of elements that are in either set. Therefore, every element in 𝐴 is also in (𝐴 ∪ 𝐵). Hence, 𝐴 is a subset of (𝐴 ∪ 𝐵).

c) 𝐴 − 𝐵 ⊆ 𝐴. This means that the difference of sets A and B is a subset of A. By definition, the difference of two sets is the set of elements that are in the first set but not in the second set. Therefore, every element in (𝐴 − 𝐵) is also in 𝐴. Hence, (𝐴 − 𝐵) is a subset of 𝐴.

d) 𝐴 ∩ (𝐵 − 𝐴) = ∅. This means that the intersection of set A and the difference of sets B and A is an empty set. By definition, the difference of two sets is the set of elements that are in the first set but not in the second set. Therefore, there are no elements in (𝐵 − 𝐴) that are also in 𝐴. Hence, the intersection of 𝐴 and (𝐵 − 𝐴) is an empty set.

e) 𝐴 ∪ (𝐵 − 𝐴) = 𝐴 ∪ 𝐵. This means that the union of set A and the difference of sets B and A is equal to the union of sets A and B. By definition, the difference of two sets is the set of elements that are in the first set but not in the second set. Therefore, the union of 𝐴 and (𝐵 − 𝐴) includes all elements that are in 𝐴 and all elements that are in 𝐵 but not in 𝐴. This is the same as the union of 𝐴 and 𝐵. Hence, 𝐴 ∪ (𝐵 − 𝐴) = 𝐴 ∪ 𝐵.