Sets and Set Operations¶

What Is a Set?¶

A set is a collection of well-defined, distinct objects (elements). Notation: curly braces.

\[ A = \{1, 2, 3\} \]

The symbol \(\in\) means "is an element of": \(2 \in A\). The symbol \(\notin\) means "is not an element of": \(5 \notin A\).

Sets can be defined by a property:

\[ B = \{x \in \mathbb{Z} : x > 0 \text{ and } x < 10\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \]

Symbol	Set
\(\emptyset\) or \(\{\}\)	Empty set (contains no element)
\(\mathbb{N}\)	Natural numbers \(\{0, 1, 2, 3, \ldots\}\)
\(\mathbb{Z}\)	Integers \(\{\ldots, -2, -1, 0, 1, 2, \ldots\}\)
\(\mathbb{Q}\)	Rational numbers
\(\mathbb{R}\)	Real numbers
\(\mathbb{C}\)	Complex numbers

\(A\) is a subset of \(B\) (\(A \subseteq B\)) if every element of \(A\) is also in \(B\):

\[ A \subseteq B \iff \forall x: x \in A \Rightarrow x \in B \]

Example. \(\{1, 3\} \subseteq \{1, 2, 3, 4\}\).

\(A \subset B\) (proper subset) means \(A \subseteq B\) and \(A \neq B\).

\[ A \cup B = \{x : x \in A \text{ or } x \in B\} \]

Example. \(\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}\).

\[ A \cap B = \{x : x \in A \text{ and } x \in B\} \]

Example. \(\{1, 2, 3\} \cap \{2, 3, 4\} = \{2, 3\}\).

\[ A \setminus B = \{x : x \in A \text{ and } x \notin B\} \]

Example. \(\{1, 2, 3\} \setminus \{2, 4\} = \{1, 3\}\).

The complement \(\overline{A}\) (or \(A^c\)) contains all elements not in \(A\) (relative to a universal set \(U\)):

\[ \overline{A} = U \setminus A \]

The power set \(\mathcal{P}(A)\) is the set of all subsets of \(A\):

\[ \mathcal{P}(\{1, 2\}) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} \]

For a set with \(n\) elements, the power set has \(2^n\) elements.

The Cartesian product \(A \times B\) is the set of all ordered pairs:

\[ A \times B = \{(a, b) : a \in A, b \in B\} \]

Example. \(\{1, 2\} \times \{a, b\} = \{(1, a), (1, b), (2, a), (2, b)\}\).