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Greetings! I've been working on basic algebra of sets.

I had really hard time to prove that identity so I've checked solution manual for this part and found this:

B ∪( A ∩ B ) = (B ∩ A) ∪ (B ∩ B ) = (B ∩ A) = A

The only case where this solution could work is when A = B (the only case that I could think of) Or am I missing something? Is there any way to prove it without A = B assumption? (I think, I could prove it using associative law, but clearly exercise only permits distributive. I am confused.)

Greetings! I've been working on basic algebra of sets.

**Refer to Exercise 2.4**. Use the identities A = A ∩ S and S = B ∪ B and a distributive law to prove that If B ⊂ A then A = B ∪ (A ∩ B). Exercise 2.4 asked to draw Venn's diagram of this case. So I did. (posted from solution manual)I had really hard time to prove that identity so I've checked solution manual for this part and found this:

B ∪( A ∩ B ) = (B ∩ A) ∪ (B ∩ B ) = (B ∩ A) = A

The only case where this solution could work is when A = B (the only case that I could think of) Or am I missing something? Is there any way to prove it without A = B assumption? (I think, I could prove it using associative law, but clearly exercise only permits distributive. I am confused.)

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