Editing Cardinality (section)

==== Combinatorics ====
{{Main|Combinatorial principles}}
[[File:Inclusion-exclusion.svg|thumb|[[Inclusion–exclusion]] illustrated for three sets.]]
[[Combinatorics]] is the area of mathematics primarily concerned with [[counting]], both as a means and as an end to obtaining results, and certain properties of finite structures. The notion cardinality of finite sets is closely tied to many basic [[combinatorial principles]], and provides a set-theoretic foundation to prove them. The above shows uniqueness of finite cardinal numbers, and therefore, <math>A \sim B</math> if and only if <math>|A| = |B|</math>, formalizing the notion of a [[bijective proof]]. 

The [[addition principle]] asserts that given [[Disjoint sets|disjoint]] sets <math>A</math> and <math>B</math>, <math>|A \cup B| = |A| + |B|</math>, intuitively meaning that the sum of parts is equal to the sum of the whole. The [[multiplication principle]] asserts that given two sets <math>A</math> and <math>B</math>, <math>|A \times B| = |A| \cdot |B|</math>, intuitively meaning that there are <math>|A| \cdot |B|</math> ways to pair objects from these sets. Both of these can be proven by a bijective proof, together with induction. The more general result is the [[inclusion–exclusion principle]], which defines how to count the number of elements in overlaping sets.