Question

3. Let A and B be sets. Prove that if A ⊆ B then (A ∪ C) ⊆ (B ∪

C).
4. Let A and B be sets. Prove that if A ⊆ B then (A − C) ⊆ (B −
C).

Answer 1

To prove that if A ⊆ B, then (A ∪ C) ⊆ (B ∪ C), we need to show that every element in (A ∪ C) is also in (B ∪ C).

Step-by-step explanation:

To prove that if A ⊆ B, then (A ∪ C) ⊆ (B ∪ C), we need to show that every element in (A ∪ C) is also in (B ∪ C).

Let x be an arbitrary element in (A ∪ C). This means that x is in either A or C (or both).

If x is in A, since A ⊆ B, x is also in B. Thus, x is in (B ∪ C).

If x is in C, then x is also in (B ∪ C) since C is a subset of (B ∪ C).

Therefore, every element in (A ∪ C) is also in (B ∪ C), which proves that if A ⊆ B, then (A ∪ C) ⊆ (B ∪ C).