A set is a well-defined collection of any kind of objects. Order of elements is not defined.
| Symbol | Description | Example |
|---|---|---|
| \(\{ \cdots \}\) | elements | \(A = \{1,2,3\}\) |
| \(\emptyset\) | empty set | \(\emptyset \equiv \{\}\) |
| \(\in\) | is member of | \(1 \in A\) |
| \(\notin\) | is not member of | \(0 \notin A\) |
| \(\lvert \cdots \rvert\) | cardinality | \(\lvert A \rvert = 3\) |
| \(\subseteq\) | subset | \(\{1, 2\} \subseteq A\) |
| \(\supseteq\) | superset | \(\{1,2,3,4\} \supseteq A\) |
| \(\{ \cdots \vert \cdots \}\) | set builder | \(\mathbb Q = \{ \frac n m \lvert n, m \in \mathbb Z , m \neq 0 \}\) |
| Symbol | Term | Definition |
|---|---|---|
| \(\cup\) | union | \(A \cup B = \{x \vert x \in A \lor x \in B \}\) |
| \(\cap\) | intersection | \(A \cap B = \{x \vert x \in A \land x \in B \}\) |
| \(-\) | set difference | \(A - B = \{x \vert x \in A \land x \notin B \}\) |
| \(\oplus\) | symetric difference | \(A \oplus B = \{x \vert (x \in A \lor x \in B) \land x \notin A \cap B \}\) |
| \(\overline{\cdots}\) | complement | \(\overline{A} = \mathbb{U} - \mathbb{A}\) |
| Symbol | Description |
|---|---|
| \(\varnothing\) | empty set |
| \(\mathbb{N}\) | natural number \(\{1,2,3,4,\ldots\}\) |
| \(\mathbb{Z}\) | integers \(\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}\) |
| \(\mathbb{Q}\) | rational numbers |
| \(\mathbb{R}\) | real numbers |
Given a set \(S\), the power set of \(S\), \(\mathcal{P}(S)\), is the set containing all possible subsets of \(S\).
\[ \begin{flalign} S &= \{a, b\} \\ \mathcal{P}(S) &= \{\varnothing , \{a\}, \{b\}, \{a,b\}\} \end{flalign} \] Cardinality of \(\mathcal{P}(S)\) is always \(2 ^ {\lvert S \rvert}\).
| Name | Union | Intersection |
|---|---|---|
| idempotent | \(A \cup A = A\) | \(A \cap A = A\) |
| identity | \(A \cup \emptyset = A\) | \(A \cap U = A\) |
| complement | \(A \cup \overline{A} = U\), \(\overline{U} = \emptyset\) | \(A \cap \overline{A} = \emptyset\), \(\overline{\emptyset} = U\) |
| domination | \(A \cup U = U\) | \(A \cap \emptyset = \emptyset\) |
| commutative | \(A \cup B = B \cup A\) | \(A \cap B = B \cap A\) |
| associative | \((A \cup B) \cup C = A \cup (B \cup C)\) | \((A \cap B) \cap C = A \cap (B \cap C)\) |
| distributive | \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\) | \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) |
| De Morgan’s laws | \(\overline{A \cup B} = \overline{A} \cap \overline{B}\) | \(\overline{A \cap B} = \overline{A} \cup \overline{B}\) |
| double complement | \(\overline{\overline{A}} = A\) | |
| absorptive | \(A \cup (A \cap B) = A\) | \(A \cap (A \cup B) = A\) |
| set difference | \(A - B = A \cap \overline{B}\) |
Two sets are equal if they contain the same elements.
Sets of the same cardinality, regardless of the elements they contain, are equivalent.
\[ \begin{gather} A = \{10, 11, 12\} \\ B = \{12, 11, 10\} \\ C = \{42, 42, 42\} \\ A = B, \quad A \neq C, \quad A \equiv C \end{gather} \]