Proof by contraposition

Proving not Q implies not P establishes the implication P implies Q.

Different ways to understand it

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Direct proof
A direct proof derives the conclusion from assumptions using justified logical steps.
Logical implication
An implication is false only when its premise is true and conclusion false.
Proof by contradiction
Assume a claim fails and derive a contradiction with established assumptions.
Axiomatic systems
An axiomatic theory derives theorems from stated axioms and inference rules.
Bijections
A bijection is both injective and surjective and has a two-sided inverse.