# Set membership

* Membership relation
* Set membership
* set elements
* set members
* subset relation
* set inclusion
* set containment
* subset
* superset
* set equality
* proper subset
* proper superset

If A = {2, 3, 4} and B = {3, 4, 5, 6}, then: A ∩ B = {3, 4} and A ∪ B = {2, 3, 4, 5, 6}. We can say that (A ∩ B) ⊂ (A ∪ B). If A = {4, 5, 6, 7} and B = {3, 4, 5, 6, 7, 8}, then A ⊂ B. If A = {4, 5, 6, 7} and B = {4, 5, 6, 7}, then A ⊆ B.

## Subset

**Set inclusion**, a binary relation derived from membership relation, is a binary relation between two sets. The relation "*is included in*", also called **subset relation**, is denoted by the symbol $$\subseteq$$.

If all members of set $$X$$ are also members of set $$Y$$, then $$X$$ is a **subset** of $$Y$$, denoted $$X\subseteq Y$$. Equivalently, $$Y$$ is a **superset** of $$X$$, denoted $$Y\supseteq X$$.

The relationship of one set being a subset of another is called **inclusion** or sometimes **containment**.

The subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

The definition for subset also means that any set is a subset (or superset) of itself.

> Any set, $$X$$, is a subset of itself, $$X\subseteq X$$

This also gives us the definition of **set equality**: two sets are equal if they containt precisely the same elements, that is, if they are each other's subsets:

> Sets $$X$$ and $$Y$$ are equal if $$X\subseteq Y$$ and $$Y\subseteq X$$.

If set $$Y$$ contains the same elements as set $$X$$, but also additional ones, then $$X$$ is a **proper subset** of $$Y$$, denoted as $$X\subset Y$$.

> $$X$$ is a proper subset of $$Y$$ if and only if $$X$$ is a subset of $$Y$$, but $$X\neq Y$$.

Set membership is not set inclusion: note that $$1,2,3$$ are elements of the set $${1,2,3}$$, but not its subsets, i.e. $$1,2,3\in {1,2,3}$$.

On the other hand, subsets, such as $${1},{2},{3}$$, are not elements of the set $${1,2,3}$$, but (some of) its subsets, e.g. $${2} \subseteq {1,2,3}$$.

## Notations

$$
let\ A={1,2,3} \ then: \ \quad 1\in A \ \quad A\ni 2 \ \quad 1,2,3\in A \ \quad {2} \not\in A \ \quad A\subseteq A \ \quad 3 \not\subseteq A \ \quad 1,2,3 \not\subseteq A \ \quad {2} \subseteq A \ \quad {1,2} \subseteq A \ \quad {1,2,3} \subseteq A \\
$$