Question

Can someone please explain step by step the following solution. i am going to post the solution i just need someone to please explain why each step is taking and what formula is being used. please explain it to me thank tyou

Suppose A is a set and {Bi | i ∈ I} is an indexed family of sets with I ≠∅.

1. Prove that A \ ( ∪i∈I Bi ) = ∩i∈I (A \ Bi).

• A be a set
• Bi | i ∈ I} is an indexed family of sets with I ≠∅
• Let x∈A - ∪i∈I Bi
• => x∈A & x∉∪i∈I Bi
• => x∈A & x∉ Bi for each i∈I
• => x∈A - Bi for each i∈I
• => x∈ ∩i∈I (A - Bi)
• so A - ( ∪i∈I Bi ) = ∩i∈I (A \ Bi) proven

2. Interpret this theorem when I = {1, 2}.

• To show that A-(B₁∪B₂)=(A-B₁) ∩(A-B₂)
• Let y∈A - (B₁∪B₂)
• => y∈A & y∉ B₁∪B₂
• => y∈A & y∉ B₁ & y∉B₂
• => y∈A & y∉ B₁ & y∈A & y∉B₂
• => y∈A - B₁ & y∈A - B₂
• => y∈ (A - B₁) ∩ y∈ (A - B₂)
• so our claim is justified and hence the proof.

Answer 1