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: \[ S = \{ 1, \pi, \text{red} \} .\]
Contents
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 \(6\). \( _\square \)