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Prove union and intersection of a set with itself equals the set

Prove that and .

Proof. Let be an arbitrary element of . Then or ; hence, . Thus, .
Conversely, if is an arbitrary element of then since it is in . Thus, .
Therefore,

Proof. Let be an arbitrary element of . Then and ; hence, . So, .
Conversely, if is arbitrary, then and ; hence, . Thus, .
Therefore,

1. Jolene Hunt says:

Hi there,

You listed Lara Alcock’s book, but misspelled her name as Laura in the link. Thanks for the recommendation though :)

2. David He says:

Hey Rori,

Should “A \cap A \subseteq A” on the second proof be reversed?

Best,

David