Logical implication

An implication is false only when its premise is true and conclusion false.

Different ways to understand it

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Proof by contraposition
Proving not Q implies not P establishes the implication P implies Q.
Propositions
A proposition has a truth value; logical connectives build compound statements.
Quantifiers
Universal and existential quantifiers specify how a statement ranges over a domain.
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.