Symbols:Set Theory

Set Delimiters
$\left\{{x, y, z}\right\}$

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

The $\LaTeX$ code for this is \left\{{x, y, z}\right\}.

Empty Set
$\varnothing$

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

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

The $\LaTeX$ code for $\varnothing$ is \varnothing, and for $\emptyset$ it is \emptyset.

Some versions of $\LaTeX$ allow \O to be used for $\emptyset$.

Set Intersection
$\cap$

"Set Intersection".

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


 * $S \cap T := \left\{{x: x \in S \land x \in T}\right\}$

Its $\LaTeX$ code 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 $T$:


 * $S \cup T := \left\{{x: x \in S \lor x \in T}\right\}$

Its $\LaTeX$ code 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.

Its $\LaTeX$ code 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 $T$.


 * $S \setminus T := \left\{{x \in S: x \notin T}\right\}$

Its $\LaTeX$ code is \setminus.

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

Cartesian Product
$\times$

The Cartesian product.

Its $\LaTeX$ code is \times</tt>.

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 $S$.

Its $\LaTeX$ code is \in</tt>.

Universal Quantifier
$\forall$

"For all".


 * $\forall x \in S: P \left({x}\right)$ means that the propositional function $P \left({x}\right)$ is true for every $x$ in the set $S$.


 * $\forall x: P \left({x}\right)$ means that the propositional function $P \left({x}\right)$ is true for every $x$ in the universal set.

Its $\LaTeX$ code is \forall</tt>.

Existential Quantifier
$\exists$

"There exists".


 * $\exists x \in S: P \left({x}\right)$ means that there exists at least one $x$ in the set $S$ for which the propositional function $P \left({x}\right)$ is true.


 * $\exists x: P \left({x}\right)$ means that there exists at least one $x$ in the universal set for which the propositional function $P \left({x}\right)$ is true.

Its $\LaTeX$ code is \exists</tt>.

Cardinality
$\left|{S}\right|$

The cardinality of the set $S$.

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

The $\LaTeX$ code for this is \left|{S}\right|</tt>.

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 $S = T$.

Its $\LaTeX$ code is \subseteq</tt>.

Proper Subset
$\subset$ or $\subsetneq$

"Proper subset".

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

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

The $\LaTeX$ code for $\subset$ is \subset</tt> and the $\LaTeX$ code for $\subsetneq$ is \subsetneq</tt>.

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 $S = T$.

Its $\LaTeX$ code is \supseteq</tt>.

Proper Superset
$\supset$ or $\supsetneq$

"Proper superset".

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

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

The $\LaTeX$ code for $\supset$ is \supset</tt> and the $\LaTeX$ code for $\supsetneq$ is \supsetneq</tt>.

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</tt> followed by the code for whatever symbol you want to negate. For example, \not \in</tt> 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:


 * $f: A \to B$: f: A \to B</tt>


 * $A \stackrel{f}{\longrightarrow} B$: A \stackrel{f}{\longrightarrow} B</tt>


 * $f: A \rightarrowtail B$: f: A \rightarrowtail B</tt>


 * $f: A \hookrightarrow B$: <tt>f: A \hookrightarrow B</tt>


 * $f: A \twoheadrightarrow B$: <tt>f: A \twoheadrightarrow B</tt>


 * $f: A \leftrightarrow B$: <tt>f: A \leftrightarrow B</tt>


 * $f: A \stackrel{\sim}{\longrightarrow} B$: <tt>f: A \stackrel{\sim}{\longrightarrow} B</tt>

= Alternative Symbols =

Set Difference
$-$

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

Its $\LaTeX$ code is <tt>-</tt>.

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.