A mathematical set is a collection of distinct objects. We denote a set using the curly braces notation \( \{ \} \). Representing sets as circles allows for a nice way to visual the interaction of a number of sets. Points inside the circle are elements of the set and points outside the circle are not elements of the set. This is commonly known as a Venn diagram.
Venn Diagram
\( A =\{1,2,3\}\), containing the elements 1,2 and 3 and the set \( B = \{2,5,6\}\) containing the elements 2,5 and 6. The sets are represented by circles.
The common element of the two sets, 2, can be seen to be in both circles.
We can generate new sets using set operations. The intersection of two sets is the set formed of the elements that are common to both and is denoted by the operator \( \cap \), and the union of two sets is the set formed of the elements that are in both and is denoted by the operator \( \cup \).
Examples of intersection and union of sets
Using the previous example \( A =\{1,2,3\}\) and \( B = \{2,5,6\}\)
\begin{align}
A \cap B = \{ 2 \} \text{ the intersection of A and B }
\end{align}
\begin{align}
A \cup B = \{ 1,2,3,5,6 \} \text{ the union of A and B }
\end{align}
Another important set operation is subtraction which is denoted by \( \backslash \) or \( – \). \( A \backslash B \) or \( A – B \) is the set of elements in \( A \) which are not in \( B \).
Example of subtraction of sets.
Let’s take \( A =\{1,2,3\}\) and \( B = \{2,5,6\}\) again. \( A\backslash B \) or \( A-B \) is the set of elements in A which are not in B.
\begin{align}
A\backslash B = \{1,3\}
\end{align}
A set \( C \) formed of some of the elements (can also be all) of another set \( A \) is a subset and is written \( C \subseteq A \). A set \( D \) which contains another set \( A \) (i.e. \( D \) is a superset of \( A \)) is written \( D \supseteq A \). Both \( \subseteq \) and \( \supseteq \) allow for equality i.e. \( A=C\) and \( A = D \).
Examples of subset and superset
Using the previous example \( A =\{1,2,3\}\) and defining \( C = \{1,2\} \) and \( D =\{1,2,3,4\}\) We have \( C \subseteq A \) and \( D \supseteq A \) since \( \{1,2\} \subseteq \{1,2,3\} \) and \( \{1,2,3,4\} \supseteq \{1,2,3\} \)
Examples of infinite sets
Let \( A \) denote the set of positive odd numbers.
$$A= \{1,3,5,7,…\}$$
Let \( B \) denote the set of even numbers \( \geq 0 \).
$$B= \{0,2,4,6,8,…\}$$
Here we define some commonly used sets in mathematics
\begin{align}
\mathbb{Z} &= \{…,-3,-2,-1,0,1,2,3,…\}\text{ (the integers)}\\
\mathbb{N} &= \{a \in \mathbb{Z}: a > 0 \} \text{ (the natural numbers)}\\
\mathbb{Q} & = \{ \frac{a}{b} : a,b \in \mathbb{Z}, b \neq 0 \} \text{ (the rational numbers)}\\
\emptyset & = \{ \} \text{ the empty set}
\end{align}
The notation \( \{x:y,z\}\) means the elements x such that conditions y and z hold.