Types of Sets

Types of Sets

## Definition / Concept

Sets are classified according to the number of elements they contain and their relationship with other sets.

## Terminologies

- Empty set → Set containing no element.

- Singleton set → Set containing exactly one element.

- Finite set → Set containing a finite number of elements.

- Infinite set → Set containing infinitely many elements.

- Equal sets → Sets containing exactly the same elements.

- Equivalent sets → Sets having the same number of elements.

## Formula

$$\varnothing=\{\}$$

$$n(\varnothing)=0$$

$$n(\{a\})=1$$

## Conditions / Special Cases

Two sets are equal if and only if:

$$A=B\Leftrightarrow A\subseteq B\text{ and }B\subseteq A$$

Two finite sets are equivalent if:

$$n(A)=n(B)$$

The empty set is different from the set containing the empty set:

$$\varnothing\ne\{\varnothing\}$$

$$n(\varnothing)=0,\qquad n(\{\varnothing\})=1$$

## Usage

- Used to classify sets according to their elements and cardinality.