A mathematical statement is a sentence with one definite truth value: true or false. A proof is a finite chain of justified steps showing that a stated conclusion must follow from the given assumptions.
An implication \(P\Rightarrow Q\) promises that whenever \(P\) is true, \(Q\) must also be true. It does not claim that \(P\) is true, and it does not automatically make the converse \(Q\Rightarrow P\) true.
| Form | Statement built from \(P\Rightarrow Q\) | Same truth value as the original? |
|---|---|---|
| Original | \(P\Rightarrow Q\) | Yes |
| Converse | \(Q\Rightarrow P\) | Not necessarily |
| Inverse | \(\neg P\Rightarrow\neg Q\) | Not necessarily |
| Contrapositive | \(\neg Q\Rightarrow\neg P\) | Yes |
The statement \(P\Rightarrow Q\) and its contrapositive \(\neg Q\Rightarrow\neg P\) are logically equivalent. This is why a proof by contraposition proves the original implication.
The quantifier “for every” makes a universal claim. Its negation is “there exists at least one value for which the claim fails”. The quantifier “there exists” makes an existence claim. Its negation is “no value has the stated property”.
| Original statement | Correct negation |
|---|---|
| For every integer \(n\), \(P(n)\) is true. | There exists an integer \(n\) for which \(P(n)\) is false. |
| There exists a real number \(x\) for which \(Q(x)\) is true. | For every real number \(x\), \(Q(x)\) is false. |
Rewrite “If \(n^2\) is divisible by \(3\), then \(n\) is divisible by \(3\)” as its converse and contrapositive. Decide which of the three statements are logically guaranteed to be equivalent.
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