Necessary and sufficient conditions

A sufficient condition implies a conclusion; a necessary condition is implied by that conclusion.

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Counterexamples
One valid counterexample disproves a universal statement but does not establish its converse.
Injective and surjective functions
Injectivity distinguishes inputs by outputs; surjectivity reaches every element of the codomain.
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.
Cartesian products
A Cartesian product consists of ordered pairs with one component from each set.