Simple Proof Methods

Fun class activities: Prove the following identities:

1. $A \cap A = A$ (Idempotent Law)
2. $A \cup A^c = U$ (Complement Law)

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Sometimes using a direct proof of a proof by cases isn't easy or isn't short and neat. For example, how do we go about proving that there exist infinitely many integers? A direct proof would most likely require us to list all the integers, but there are infinitely many of them! So what should we do?

If a direct proof doesn't seem appropriate, we'll resort to a proof by contradiction: this kind of proof assumes that a given statement (e.g., "there exist infinitely many integers") is false and then derives a contradiction, proving that the assumption we made must have been incorrect. That is, the original statement is, therefore, true.