Combinatorics

combinatorics

Definition

Combinatorics

Combinatorics is a branch of discrete mathematics that deals with combinations of objects to a finite set in accordance with certain constraints.

Rules

Combinatorics is based on the following rules:

Sum Rule

Let $A$ and $B$ be two finite sets, then:

$$A\cap B\ne \varnothing ⟹|A\cup B|=|A|+|B|$$

Product Rule

Let $A$ and $B$ be two finite sets, then:

$$|A\times B|=|A|\times |B|$$

Equality Rule

If there exists a bijective relation between two finite sets $A,B$, then $A$ has as many elements as $B$:

$$\exists f:A\to B\text{with f bijective}⟹|A|=|B|$$