Sets - Elements

Elements are the objects contained in a set. A set may be defined by a common property amongst the objects. For example, the set $E$ of positive even integers is the set $$E = \{ 2, 4, 6, 8, 10 \ldots \} .$$

The set $F$ of living people is the set $$F = \{\text{Steve Buscemi}, \text{Jesse Jackson}, \cdots\}.$$

A set can also be defined by simply stating its elements. For example, one can define the set $S$ by writing its elements, as follows: $$S = \{ 1, \pi, \text{red} \} .$$

The mathematical notation for "is an element of" is $\in$. For example, to denote that $2$ is an element of the set $E$ of positive even integers, one writes $2 \in E$ . To indicate that an element, 3, is not in the set $E$, write 3 $\notin E$.

The size of a set (also called its cardinality) is the number of elements in the set. For example, the size of the set $\{2, 4, 6 \}$ is $3,$ while the size of the set $E$ of positive even integers is infinity.

What is the size of the set $\{ 1, 3, 5, 7, 9, 11 \}$?

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The elements in the set are $1, 3, 5, 7, 9$, and $11.$ There are $6$ of them. Hence, the size of the set is $6$. $_\square$