Sets

Description

A set is an unordered collection of distinct objects, called elements or members of the set. A set is said to contain its elements. We write \(a \in A\) to denote that a is an element of the set \(A\). The notation \(a \notin A\) denotes that a is not an element of the set \(A\).

Popular sets:

• \(\mathbb{N} = \{ 0, 1, 2, 3, \ldots \}\), the set of all natural numbers
• \(\mathbb{Z} = \{ \ldots, -2, -1, 0, 1, 2, \ldots \}\), the set of all integers
• \(\mathbb{Z}^+ = \{ 1, 2, 3, \ldots \}\), the set of all positive integers
• \(\mathbb{Q} = \left\{ \frac{p}{q} \ \middle| \ p \in \mathbb{Z}, \ q \in \mathbb{Z}, \ \text{and} \ q \neq 0 \right\}\), the set of all rational numbers
• \(\mathbb{R}\), the set of all real numbers
• \(\mathbb{R}^+\), the set of all positive real numbers
• \(\mathbb{C}\), is the set of all complex numbers

Subsets

The set \(A\) is a subset of \(B\), and \(B\) is a superset of \(A\), if and only if every element of \(A\) is also an element of \(B\). We use the notation \(A \subseteq B\) to indicate that \(A\) is a subset of the set \(B\). If, instead, we want to stress that \(B\) is a superset of \(A\), we use the equivalent notation \(B \supseteq A\).(So, \(A \subseteq B\) and \(B \supseteq A\) are equivalent statements.)

Operations

Cartesian Products

Let \(A\) and \(B\) be sets. The Cartesian product of \(A\) and \(B\), denoted by \(A \times B\), is the set of all ordered pairs \((a, b)\), where \(a \in A\) and \(b \in B\). Hence,

\(A \times B = \{ (a, b) \mid a \in A \land b \in B \}\)

Union

Let \(A\) and \(B\) be sets. The union of the sets \(A\) and \(B\), denoted by \(A \cup B\), is the set that contains those elements that are either in \(A\) or in \(B\), or in both.

Intersection

Let \(A\) and \(B\) be sets. The intersection of the sets \(A\) and \(B\), denoted by \(A \cap B\), is the set containing those elements in both \(A\) and \(B\).

Difference

Let \(A\) and \(B\) be sets. The difference of \(A\) and \(B\), denoted by \(A - B\), is the set containing those elements that are in \(A\) but not in \(B\). The difference of \(A\) and \(B\) is also called the complement of B with respect to \(A\).

Complement

Let \(U\) be the universal set. The complement of the set \(A\), denoted by \(-A\), is the complement of \(A\) with respect to \(U\). Therefore, the complement of the set \(-A\) is \(U - A\)

Identities