Injective and surjective functions

Injectivity distinguishes inputs by outputs; surjectivity reaches every element of the codomain.

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Bijections
A bijection is both injective and surjective and has a two-sided inverse.
Necessary and sufficient conditions
A sufficient condition implies a conclusion; a necessary condition is implied by that conclusion.
Axiomatic systems
An axiomatic theory derives theorems from stated axioms and inference rules.
Cartesian products
A Cartesian product consists of ordered pairs with one component from each set.
Countability
A set is countable if it is finite or can be enumerated by natural numbers.