Sets
Set
A set is a collection of things.
The things are called elements of a set.
\[ Colors = \{Red, Blue, Green\} \]
\[ Numbers = \{1,2,3\} \]
Sets can be finite or infinite.
\[ Some\ Even\ Integers = \{2,4,6\} \]
\[ All\ Even\ Integers = \{..., -4, -2, 0, 2, 4, ...\} \]
The number of elements in a set is called the cardinality.
\[ |Colors| = 3 \]
Two sets are equal if they share exactly the same elements.
\[ A = \{2,4,6\}, B = \{4,2,6\}, C = \{4,2,7\} \]
\[ A = B \] \[ A \neq C \]
To express that \(2\) is an element of \(A\), we denote:
\[ 2 \in A \] \[ \text{2 exists in A} \]
\[ 5 \notin A \] \[ \text{5 does not exist in A} \]
Some sets are so significant that we reserve special symbols for them:
\[ \emptyset = \{\} \quad \textbf{(empty set)} \]
\[ \mathbb{N} = \{1, 2, 3, ... \} \quad \textbf{(natural numbers)} \]
\[ \mathbb{Z} = \{..., -2, -1, 0, 1, 2, ... \} \quad \textbf{(integers)} \]
\[ \mathbb{R} = \{..., -0.22,...,0,...,1,..., \pi, ... \} \quad \textbf{(real numbers)} \]
We visualize \(\mathbb{R}\) as an infinitely long number line.
Set-Builder Notation
A special notation called set-builder notation is used to describe sets that are too big or complex to list between braces.
\[ All\ Even\ Integers_{1} = \{..., -4, -2, 0, 2, 4, ...\} \]
\[ All\ Even\ Integers_{2} = \{2x: x \in \mathbb{Z} \} \]
\[ \text{The set of all numbers of the form } 2x \text{ such that } x \in \mathbb{Z} \]
\[ All\ Even\ Integers_{1} = All\ Even\ Integers_{2} \]
Exercise
Write the following sets in set-builder notation:
- \(\{ 2, 4, 8, 16, 32, 64, ... \}\)
- \(\{ 0, 1, 4, 9, 16, 25, 36, ... \}\)
- \(\{ 3, 4, 5, 6, 7, 8 \}\)
- \(\{ 2^x: x \in \mathbb{N} \}\)
- \(\{ x^2: x \in \mathbb{Z} \}\)
- \(\{ x \in \mathbb{Z}: 3 \le x \le 8 \}\)