Proof by exhaustion
Brute-force by nature, and not particularly clever.
A proof by exhaustion is a type of direct proof in which a theorem is shown to be true by checking each element in its domain individually. Proofs by exhaustion are only used to prove a universal statement when the domain is small. If the domain is large or infinite, you'll have to use universal generalization instead.
Here's an example of a proof by exhaustion:
Theorem: For every \(n \in \{2, 4, 6\}\), \(n\) is even.
Proof by exhaustion: Let's check the theorem for every element in \(\{2, 4, 6\}\): $$n=2:2=2(1)\checkmark$$ $$n=4:4=2(2)\checkmark$$ $$n=6:6=2(3)\checkmark$$ Thus, for every \(n \in \{2, 4, 6\}\), \(n\) is even. \(■\)
Proof by exhaustion: Let's check the theorem for every element in \(\{2, 4, 6\}\): $$n=2:2=2(1)\checkmark$$ $$n=4:4=2(2)\checkmark$$ $$n=6:6=2(3)\checkmark$$ Thus, for every \(n \in \{2, 4, 6\}\), \(n\) is even. \(■\)
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