Counterexamples

One valid counterexample disproves a universal statement but does not establish its converse.

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Axiomatic systems
An axiomatic theory derives theorems from stated axioms and inference rules.
Necessary and sufficient conditions
A sufficient condition implies a conclusion; a necessary condition is implied by that conclusion.
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.
Countability
A set is countable if it is finite or can be enumerated by natural numbers.