1.3 Element Inclusion and Truth Values

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 1.1 Set

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Example

Let's define the set of vowels.

\begin{aligned} L_{v} = {a, e, i, o, u} \\ U_{v} = {A, E, I, O, U} \end{aligned}

In set notation, brackets have very special meaning.

${$ left bracket
$}$ right bracket

Additionally, in set notation, the order of elements doesn't matter.

\begin{aligned} L_{v} & = {a, e, i, o, u} \\ = {u, o, i, e, a} \\ = {i, o, a, u, e} \end{aligned}

The number of appearances doesn't matter either.

L_{v} = {e, i, e, i, o, a, u}

The only defining characteristic that matters is whether or not an element is included in the set (aka set inclusion). To denote inclusion, we use the set membership symbol.

$x \in A$ is true if element $x$ is inside set $A$
$x \in A$ is false if $x$ is not in set $A$ (in which case $x \notin A$ is true)