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Using the contrapositive begins with $n=2k+1$, which gives us very clear and usable information about $n$. Perhaps you could put a heuristic in the following form: "try both ways, just for a couple of steps, and see if either looks notably easier than the other".
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.
We are now able to use contradiction and contrapositive to prove two classical theorems in mathematics. In Activity 3.4.2 you used contrapositive to prove if \(n^2\) is even, then \(n\) is even. This statement is an important step in our first big theorem.
Exercise 16.1 Use the following examples to practise proof by contrapositive. Consider why this method is easier than a direct proof for these conjectures. Conjecture 16.1 : If \ (a^2+b^2\) is odd and \ (a\) and \ (b\) are both integers, then \ (a\) or \ (b\) have different parity to one another.
Procedure \(\PageIndex{1}\): Proof by proving the contrapositive. To prove \(P \Rightarrow Q\text{,}\) you can instead prove \(\neg Q \Rightarrow \neg P\text{.}\)
To prove $P \rightarrow Q$, you can do the following: Prove directly, that is assume $P$ and show $Q$; Prove by contradiction, that is assume $P$ and $\lnot Q$ and derive a contradiction; or. Prove the contrapositive, that is assume $\lnot Q$ and show $\lnot P$.
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 \(p\).
