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In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive.
Example \(\PageIndex{2}\) Prove that every prime number larger than \(2\) is odd. Solution. We want to prove the following universally quantified conditional (“for all \(p\)” omitted, domain is positive integers).
21 lis 2023 · The law of contrapositive states that the original statement is true if, and only if, the contrapositive is true. If the contrapositive is false, the original statement is false.
28 lis 2020 · contrapositive: If a conditional statement is \(p\rightarrow q\) (if \(p\) then q), then the contrapositive is \(\sim q\rightarrow \sim p\) (if not q then not p). converse: If a conditional statement is \(p\rightarrow q\) (if \(p\), then \(q\)), then the converse is \(q\rightarrow p\) (if \(q\), then \(p\).
1 What is a Contrapositive? A Counter-Example? A Converse? \if P then Q" is logically equivalent to \if not Q then not P" Our goal is to get to the point where we can do the contrapositive mentally. In other words, we want to be able to read a conditional statement (if P then Q) and immediately \see" the contrapositive. There are
11 lis 2022 · The contrapositive of an implication is the converse of its inverse (or the inverse of its converse, which amounts to the same thing). That is, \[\text{ the contrapositive of } A\Rightarrow B\text{ is the implication }\lnot B\Rightarrow\lnot A\]
Proof. From the map, it’s easy to see the contrapositive of the conjecture is “If \ (a, b\) both odd or both even, then \ (a^2+b^2\) is even.”. Case 1:\ (a\), \ (b\) both odd. We have \ [\begin {align} a &= 2k +1 \\ b &= 2l + 1 \end {align}\] where \ (k,l \in \mathbb Z\).
