# Symbols:Set Theory

## Symbols used in Set Theory

### Set Delimiters

$\set {x, y, z}$

Denotes that the objects $x, y, z$ are the elements of a set.

The $\LaTeX$ code for $\set {x, y, z}$ is \set {x, y, z} .

### Is an Element Of

$\in$

Let $S$ be a set.

An element of $S$ is a member of $S$.

The $\LaTeX$ code for $x \in S$ is x \in S .

### Contains as an Element

$\ni$

$S \ni x$ means that $x$ is an element of the set $S$.

The $\LaTeX$ code for $S \ni x$ is S \ni x .

### Subset

$\subseteq$

$S \subseteq T$ means $S$ is a subset of $T$.

In other words, every element of $S$ is also an element of $T$.

Note that this symbol allows the possibility that $S = T$.

The $\LaTeX$ code for $\subseteq$ is \subseteq .

### Proper Subset

$\subsetneq$ or $\subsetneqq$

$S \subsetneq T$ and $S \subsetneqq T$ both mean:

$S$ is a proper subset of $T$

In other words, $S \subseteq T$ and $S \ne T$.

The $\LaTeX$ code for $\subsetneq$ is \subsetneq .

The $\LaTeX$ code for $\subsetneqq$ is \subsetneqq .

### Superset

$\supseteq$

$S \supseteq T$ means $S$ is a superset of $T$, or equivalently, $T$ is a subset of $S$.

Thus every element of $T$ is also an element of $S$.

Note that this symbol allows the possibility that $S = T$.

The $\LaTeX$ code for $\supseteq$ is \supseteq .

### Proper Superset

$\supsetneq$ or $\supsetneqq$

$S \supsetneq T$ and $S \subsetneqq T$ both mean:

$S$ is a proper superset of $T$

In other words, $S \supseteq T$ and $S \ne T$.

The $\LaTeX$ code for $\supsetneq$ is \supsetneq .

The $\LaTeX$ code for $\supsetneqq$ is \supsetneqq .

### Set Intersection

$\cap$

$S \cap T$ denotes the intersection of $S$ and $T$.

That is, $S \cap T$ is defined to be the set containing all the elements that are in both the sets $S$ and $T$:

$S \cap T := \set {x: x \in S \land x \in T}$

The $\LaTeX$ code for $\cap$ is \cap .

### Set Union

$\cup$

Let $S$ and $T$ be sets.

The (set) union of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$:

$x \in S \cup T \iff x \in S \lor x \in T$

The $\LaTeX$ code for $\cup$ is \cup .

### Empty Set

$\O$

The empty set:

$\O = \set {}$

The $\LaTeX$ code for $\O$ is \O  or \varnothing or \empty.

### Deprecated

An alternative but less attractive symbol for the empty set is $\emptyset$.

This symbol is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

The $\LaTeX$ code for $\emptyset$ is \emptyset .

### Set Difference

$\setminus$

The difference between two sets $S$ and $T$ is denoted $S \setminus T$ and consists of all the elements of $S$ which are not elements of $T$.

$S \setminus T := \set {x \in S: x \notin T}$

The $\LaTeX$ code for $\setminus$ is \setminus .

### Symmetric Difference

$\symdif$

The symmetric difference between two sets $S$ and $T$ is denoted $S \symdif T$ and consists of all the elements in either set, but not in both.

$S \symdif T := \paren {S \setminus T} \cup \paren {T \setminus S}$

The $\LaTeX$ code for $\symdif$ is \symdif .

### Cartesian Product

$\times$

$S \times T$ denotes the Cartesian product of $S$ and $T$.

The $\LaTeX$ code for $S \times T$ is S \times T .

### Order Sum

$\oplus$

$S_1 \oplus S_2$ denotes the order sum of two ordered sets $S_1$ and $S_2$.

The $\LaTeX$ code for $\oplus$ is \oplus .

### Cardinality

$\card S$

The cardinality of the set $S$.

For finite sets, this means the number of elements in $S$.

The $\LaTeX$ code for $\card {S}$ is \card {S} .

### Set Equivalence

$S \sim T$

Let $S$ and $T$ be sets.

Then $S$ and $T$ are equivalent if and only if:

there exists a bijection $f: S \to T$ between the elements of $S$ and those of $T$.

That is, if they have the same cardinality.

This can be written $S \sim T$.

If $S$ and $T$ are not equivalent we write $S \nsim T$.

The $\LaTeX$ code for $S \sim T$ is S \sim T .

The $\LaTeX$ code for $S \nsim T$ is S \nsim T .

### Mapping

A mapping $f \subset A \times B$ can be written:

$f: A \to B$

or:

$A \stackrel {f} {\longrightarrow} B$

If $a \in A$ and $b \in B$ such that $\map f a = b$ then we can write:

$f: a \mapsto b$

If $f$ is an injection this can be written:

$f: A \rightarrowtail B$ or $f: A \hookrightarrow B$

Similarly a surjection can be written:

$f: A \twoheadrightarrow B$

Notations for bijection include:

$f: A \leftrightarrow B$ or $f: A \stackrel {\sim} {\longrightarrow} B$

The $\LaTeX$ code for these symbols are as follows:

The $\LaTeX$ code for $f: A \to B$ is f: A \to B .
The $\LaTeX$ code for $A \stackrel {f} {\longrightarrow} B$ is A \stackrel {f} {\longrightarrow} B .
The $\LaTeX$ code for $f: a \mapsto b$ is f: a \mapsto b .
The $\LaTeX$ code for $f: A \rightarrowtail B$ is f: A \rightarrowtail B .
The $\LaTeX$ code for $f: A \hookrightarrow B$ is f: A \hookrightarrow B .
The $\LaTeX$ code for $f: A \twoheadrightarrow B$ is f: A \twoheadrightarrow B .
The $\LaTeX$ code for $f: A \leftrightarrow B$ is f: A \leftrightarrow B .
The $\LaTeX$ code for $f: A \stackrel {\sim} {\longrightarrow} B$ is f: A \stackrel {\sim} {\longrightarrow} B .

### Composition of Mappings

$\circ$

The composite mapping $f_2 \circ f_1$ is defined as:

$\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$

The $\LaTeX$ code for $f \circ g$ is f \circ g .

### Inverse Mapping

$f^{-1}$

Let $f: S \to T$ be a mapping.

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$:

$f^{-1} := \set {\tuple {t, s}: \map f s = t}$

Let $f^{-1}$ itself be a mapping:

$\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$

and

$\forall y \in T: \exists x \in S: \tuple {y, x} \in f$

Then $f^{-1}$ is called the inverse mapping of $f$.

The $\LaTeX$ code for $f^{-1}$ is f^{-1} .

### Universal Quantifier

$\forall$
$\forall x \in S: \map P x$ means that the propositional function $\map P x$ is true for every $x$ in the set $S$.
$\forall x: \map P x$ means that the propositional function $\map P x$ is true for every $x$ in the universal set.

The $\LaTeX$ code for $\forall$ is \forall .

### Existential Quantifier

$\exists$
$\exists x \in S: \map P x$ means that there exists at least one $x$ in the set $S$ for which the propositional function $\map P x$ is true.
$\exists x: \map P x$ means that there exists at least one $x$ in the universal set for which the propositional function $\map P x$ is true.

The $\LaTeX$ code for $\exists$ is \exists .

### Negation

$\not \in, \not \exists, \not \subseteq, \not \subset, \not \supseteq, \not \supset$

The above symbols all mean the opposite of the non struck through version of the symbol.

For example, $x \not \in S$ means that $x$ is not an element of $S$.

The slash through a symbol ($/$) can be used to reverse the meaning of essentially any mathematical symbol (especially relations), although it is used most frequently with those listed above.

Note that $\not \subsetneq$ and $\not \supsetneq$ can be confusing due to the strike through of the symbol as a whole and the strike through of the equivalence bar on the bottom, and hence should likely be avoided.

The $\LaTeX$ code for negation is \not followed by the code for whatever symbol you want to negate.

For example, \not \in will render $\not\in$.

## Variants

### Set Difference

$-$

A variant notation for the difference between two sets $S$ and $T$ is $S - T$.

The $\LaTeX$ code for $S - T$ is S - T .

### Symmetric Difference

There is no standard symbol for symmetric difference. The one used here, and in general on $\mathsf{Pr} \infty \mathsf{fWiki}$:

$S \symdif T$

is the one used in 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics.

The following are often found for $S \symdif T$:

$S * T$
$S \oplus T$
$S + T$
$S \mathop \triangle T$

According to 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics: symmetric difference:

$S \mathop \Theta T$
$S \mathop \triangledown T$

are also variants for denoting this concept.

## Deprecated Symbols

This page contains symbols which may or may not be in current use, but are either non-standard in mathematics or have been superseded by their more modern variants.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the intention is to present a consistent style, and so these symbols are to be considered deprecated.

### Subset

$\subset$ is sometimes used to mean:

$S$ is a subset of $T$

in the sense that $S$ is permitted to equal $T$.

That is, for which we have specified as $S \subseteq T$.

Although many sources use this interpretation, it is emphatically not recommended, as it can be the cause of considerable confusion.

### Superset

$\supset$ is sometimes used to mean:

$S$ is a superset of $T$

in the sense that $S$ is permitted to equal $T$.

That is, for which we have specified as $S \supseteq T$.

Although many sources use this interpretation, it is emphatically not recommended, as it can be the cause of considerable confusion.