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

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