Finite set

A set is finite if its elements can be put in bijection with a natural number, including zero under the convention that zero is natural.

Set: A collection of distinct objects treated as a single mathematical object. Finiteness is a property of sets, defined through their elements.
Pigeonhole principle: A counting principle stating that placing more objects than containers forces at least one container to hold multiple objects. It follows from the impossibility of injecting a larger finite set into a smaller one.
Infinite set: A set that is not in bijection with any natural-number set. It is the direct opposite of a finite set.
Combinatorics: The branch of mathematics concerned with counting, arranging, and selecting discrete structures. Its basic problems ask for the sizes and arrangements of finite sets.
Cardinality: The number of elements in a set, or the size assigned to a more general set. A set is finite exactly when its cardinality is a natural number.
Finite union: The union of finitely many sets, formed by including every element that belongs to at least one of them. A finite union of finite sets remains finite.
Countably infinite set: An infinite set whose elements can be listed in a sequence indexed by the natural numbers. It is infinite despite being listable, unlike a finite set.
Graph theory: The study of graphs, structures consisting of vertices joined by edges. Finite graphs are built from finite vertex and edge sets.
Bijection: A function that is both injective and surjective, pairing every element of its domain with exactly one element of its codomain. A bijection with a natural-number set witnesses finiteness.
Cartesian product: The set of ordered tuples formed by choosing one element from each of several sets. Products of finitely many finite sets are finite, with sizes multiplied.

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