1. Prove that .
Let . Then and . Therefore, and , so
In particular, and , so , and hence .
Conversely, suppose . Then , so
Now , so combining this with gives . Likewise, , and combining this with gives . All together,
Thus, and , so and . Consequently, .
Since and , it follows that .
2. Let A, B, and C be sets. Prove the following statement using the definitions of union, intersection, and complement and the rules of logic. Justify each step.
We'd be wise to question why we hold a grudge as if it were going to make us happy and ease our pain. It's rather like eating rat poison and thinking the rat will die. - Pema Ch\"odr\"on
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