# 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: \[ S = \{ 1, \pi, \text{red} \} .\]

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## Elements of a Set

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 \(\notin E\).

Here is a set containing all of the players on a volleyball team.

\(Team = \{\text{John}, \text{Ashley}, \text{Lisa}, \text{Joe}\}\).

Which one of the choices below is not true?

\(\text{Ashley} \in Team\)

\(\text{John} \notin Team\)

\(\text{Adam} \notin Team\)

\(\text{Joe} \in Team\)

## The Size of a Set

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 \( \{ 1, 3, 5, 7, 9, 11 \} \)?

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 \(6\). \( _\square \)