# Symbols:Set Operations and Relations

## 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} .

## Empty Set

$\O$

The empty set: $\O = \set {}$.

An alternative but less attractive symbol for the same thing is $\emptyset$.

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

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

## Set Intersection

$\cap$

$S \cap T$ 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

$\cup$

"Set Union".

$S \cup T$ is defined to be the set containing all the elements that are in either or both of the sets $S$ and $T$:

$S \cup T := \set {x: x \in S \lor x \in T}$

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

## Ordered Sum

$+$

$S_1 + S_2$ denotes the ordered sum of two sets $S_1$ and $S_2$.

See Arithmetic and Algebra and Abstract Algebra for alternative definitions of this symbol.

The $\LaTeX$ code for $+$ is + .

## 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 .

See Number Theory: Divisor for an alternative use of this symbol.

## Cartesian Product

$\times$

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

See Arithmetic and Algebra and Vector Algebra for alternative definitions of this symbol.

## Is an Element Of

$\in$

"Element of". $x \in S$ means that $x$ is an element of the set $S$.

The $\LaTeX$ code for $\in$ is \in .

## 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 .

## Universal Quantifier

$\forall$

"For all".

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

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

## Existential Quantifier

$\exists$

"There exists".

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

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

## 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} .

See Arithmetic and Algebra, Complex Analysis and Abstract Algebra for alternative definitions of this symbol.

## Subset

$\subseteq$

"Subset".

$S \subseteq T$ 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

$\subset$, $\subsetneq$ or $\subsetneqq$

$S \subset T$ means "$S$ is a proper subset of $T$", in other words, $S \subseteq T$ and $S \ne T$.

The symbols $\subset$, $\subsetneq$ and $\subsetneqq$ are equivalent.

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

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

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

## Superset

$\supseteq$

"Superset".

$S \supseteq T$ 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

$\supset$, $\supsetneq$ or $\supsetneqq$

$S \supset T$ means $S$ is a proper superset of $T$, in other words, $S \supseteq T$ and $S \ne T$.

The symbols $\supset$, $\supsetneq$ and $\supsetneqq$ are equivalent.

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

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

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

It should be noted that use in the literature of subset-type symbols is haphazard: many authors use exclusively $\supset$, even when the inclusion is not strict, reserving $\supsetneq$ or $\supsetneqq$ for strict inclusions. If in doubt, one cannot go wrong by writing $\supseteq$, the reader can then consider it an ongoing exercise to determine which inclusions are strict.

## Negation

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

"Negation".

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 $\not\in$.

## Mappings

A mapping $f \subset A \times B$ is usually written:

$f: A \to B$ or $A \stackrel {f} {\longrightarrow} B$

If $f$ is injective sometimes this is written:

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

Similarly surjectivity 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 $f: A \rightarrowtail B$ 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 .

## Alternative Symbols

### Set Difference

$-$

An alternative notation for the difference between two sets $S$ and $T$ is $S - T$.

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

See Arithmetic and Algebra and Logical Operators for alternative definitions of this symbol.

## Deprecated Symbols

### Subset, Superset

$\subset$ 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 $S \subseteq T$.

Similarly, $\supset$ is sometimes used to mean $S \supseteq T$.

Although many sources use these interpretations, they are emphatically not recommended, as they can be the cause of considerable confusion.