Let $X$ be a set of people. Define the binary relation $R \subseteq X \times X$ by

R = {(a, b) \in X \times X ∣ a and b are friends} .

This relation $R$ satisfies the following properties: \begin{itemize} \item \textbf{Reflexive:} $\forall a \in X, (a, a) \in R$ (everyone is their own friend), \item \textbf{Symmetric:} $\forall a, b \in X, (a, b) \in R \Rightarrow (b, a) \in R$ , \item \textbf{Not necessarily transitive:} $(a, b) \in R \land (b, c) \in R ⇏ (a, c) \in R$ . \end{itemize}

Let $X$ be a set of people. Define the binary relation $R \subseteq X \times X$ such that for $a, b \in X$ ,

(a, b) \in R ⟺ a and b are friends .

Set Theory Notation Reference

Basic Set Relations

Element	Definition	Description
$\in$	$x \in A$	” $x$ is an element of set $A$ "
$\in /$	$x \in / A$	" $x$ is not an element of set $A$ "
$\subseteq$	$A \subseteq B ⟺ \forall x (x \in A ⟹ x \in B)$	" $A$ is a subset of $B$ ” (all elements of $A$ are in $B$ )
$\subset$	$A \subset B ⟺ A \subseteq B \land A \neq = B$	” $A$ is a proper subset of $B$ ” (subset but not equal)
$\supseteq$	$A \supseteq B ⟺ B \subseteq A$	” $A$ is a superset of $B$ "
$\supset$	$A \supset B ⟺ B \subset A$	" $A$ is a proper superset of $B$ "
$=$	$A = B ⟺ \forall x (x \in A ⟺ x \in B)$	" $A$ equals $B$ ” (same elements)
$\neq =$	$A \neq = B ⟺ \neg (A = B)$	” $A$ does not equal $B$ “

Set Operations

Element	Definition	Description
$\cup$	$A \cup B = x : x \in A \lor x \in B$	Union: elements in either $A$ or $B$ or both
$\cap$	$A \cap B = x : x \in A \land x \in B$	Intersection: elements in both $A$ and $B$
$∖$	$A ∖ B = x : x \in A \land x \in / B$ Also $A - B$	Set difference: elements in $A$ but not in $B$
$△$	$A △ B = (A ∖ B) \cup (B ∖ A)$	Symmetric difference: elements in $A$ or $B$ but not both
$A^{c}$	$A^{c} = x \in U : x \in / A$	Complement of $A$ (relative to universal set $U$ )
$\times$	$A \times B = (a, b) : a \in A \land b \in B$	Cartesian product: set of ordered pairs

Special Sets

Element	Definition	Description
$\emptyset$	$\emptyset = x : x \neq = x$	Empty set (set with no elements)
$U$	Context-dependent	Universal set (contains all elements under consideration)
$N$	$N = 1, 2, 3, \dots$	Natural numbers (positive integers)
$Z$	$Z = \dots, - 2, - 1, 0, 1, 2, \dots$	Integers
$Q$	$Q = \frac{p}{q} : p, q \in Z, q \neq = 0$	Rational numbers
$R$	All real numbers	Real numbers (includes rationals and irrationals)
$C$	$C = a + bi : a, b \in R, i^{2} = - 1$	Complex numbers

Set Construction

Element	Definition	Description
$a, b, c$	Explicit enumeration	Set containing elements $a$ , $b$ , $c$
$x : P (x)$	$x : P (x) = x ∣ P (x) is true$	Set of all $x$ such that property $P (x)$ holds
$x ∣ P (x)$	Alternative notation for set-builder	Same as above
$[a, b]$	$[a, b] = x \in R : a \leq x \leq b$	Closed interval from $a$ to $b$
$(a, b)$	$(a, b) = x \in R : a < x < b$	Open interval from $a$ to $b$
$[a, b)$	$[a, b) = x \in R : a \leq x < b$	Half-open interval (closed at $a$ , open at $b$ )
$(a, b]$	$(a, b] = x \in R : a < x \leq b$	Half-open interval (open at $a$ , closed at $b$ )

Cardinality and Size

Element	Definition	Description
$∥ A ∥$	$∥ A ∥ =$ number of elements in $A$	Cardinality of finite set $A$
$ℵ_{0}$	$∥ N ∥ = ℵ_{0}$	Cardinality of countably infinite sets
$2^{A}$	$2^{A} = S : S \subseteq A$	Power set of $A$ (set of all subsets)
$P (A)$	$P (A) = 2^{A}$	Alternative notation for power set

Advanced Notations

Element	Definition	Description
$⋃_{i \in I} A_{i}$	$⋃_{i \in I} A_{i} = x : \exists i \in I, x \in A_{i}$	Union over indexed collection
$⋂_{i \in I} A_{i}$	$⋂_{i \in I} A_{i} = x : \forall i \in I, x \in A_{i}$	Intersection over indexed collection
$\sim$	$a \sim b$ if relation holds	Equivalence relation
$\propto$	$f (x) \propto g (x)$ if $f (x) = c \cdot g (x)$	Proportional to (for some constant $c$ )

Common Examples

Set Difference Example: If $A = 1, 2, 3, 4, 5$ and $B = 3, 4, 5, 6, 7$ , then:

$A ∖ B = 1, 2$
$B ∖ A = 6, 7$

Set Builder Notation:

$x \in R : x^{2} < 4 = (- 2, 2)$
$n \in N : n is even = 2, 4, 6, 8, \dots$

Quartz 4

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2025-06-20-sets