Question

Prove that if A_1, A_2,...,A_n and B_1, B_2,...,B_n are sets

such that A_j ⊆ B_j for j = 1, 2,...,n, then
∪_(j = 1)^n A_j ⊆ ∪_(j = 1)^n B_j .

Answer 1

To prove that if A1, A2,...,An and B1, B2,...,Bn are sets such that Aj ⊆ Bj for j = 1, 2,...,n, then ∪j = 1n Aj ⊆ ∪j = 1n Bj, we need to show that if an element x belongs to the union of sets Aj, then it also belongs to the union of sets Bj.

Step-by-step explanation:

To prove that if A1, A2,...,An and B1, B2,...,Bn are sets such that Aj ⊆ Bj for j = 1, 2,...,n, then ∪j = 1n Aj ⊆ ∪j = 1n Bj, we need to show that if an element x belongs to the union of sets Aj, then it also belongs to the union of sets Bj.

1. Let x be an element that belongs to ∪j = 1n Aj.
2. Since x belongs to the union, there exists an index k such that x belongs to set Ak.
3. Since Ak ⊆ Bk, x also belongs to set Bk.
4. Since there exists an index k for which x belongs to set Bk, x belongs to the union of sets Bj.

Therefore, if x belongs to the union of sets Aj, then x also belongs to the union of sets Bj. This proves that ∪j = 1n Aj ⊆ ∪j = 1n Bj.