Subset and Proper Subset

Subset and Proper Subset

## Definition / Concept

A set $A$ is a subset of $B$ if every element of $A$ also belongs to $B$.

## Formula

$$A\subseteq B\Leftrightarrow(\forall x\in A,\ x\in B)$$

A proper subset satisfies:

$$A\subset B\Leftrightarrow A\subseteq B\text{ and }A\ne B$$

## Conditions / Special Cases

$$\varnothing\subseteq A$$

$$A\subseteq A$$

If:

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

then:

$$A=B$$

## Usage

- Used to establish equality of sets.

- Used in power-set and counting problems.