Cardinality of a Set

Cardinality of a Set

## Definition / Concept

The cardinality of a finite set is the number of distinct elements present in it.

## Formula

$$n(A)=\text{Number of distinct elements in }A$$

For two finite sets:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

For three finite sets:

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$$

## Terminologies

- Cardinality → Number of elements in a set.

- Disjoint sets → Sets having no common element.

## Conditions / Special Cases

If:

$$A\cap B=\varnothing$$

then:

$$n(A\cup B)=n(A)+n(B)$$

If $A$, $B$ and $C$ are pairwise disjoint:

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)$$

## Usage

- Used in counting problems involving overlapping or disjoint sets.