Symbols:Set Theory

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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 symbol universally used in modern mainstream mathematics to mean $x$ is an element of $S$ is:

$x \in S$

Similarly, $x \notin S$ means $x$ is not an element 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

$\cup$

$S \cup T$ denotes the union of $S$ and $T$.


That is, $S \cup T$ 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 .


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 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

$*$

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

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


The $\LaTeX$ code for \(\ast\) is \ast .


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 .


Ordered Sum

$+$

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


The $\LaTeX$ code for \(+\) is + .


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} .


Mapping

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

$f: A \to B$

or:

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


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 \(f: A \rightarrowtail B\) 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 .


Universal Quantifier

$\forall$

For all.

$\forall x \in S: \map P x$ 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

$\exists$

There 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 $\mathsf{Pr} \infty \mathsf{fWiki}$:

$S * T$

is the one used in 1971: Allan Clark: Elements of Abstract Algebra.


The following are often found for $S * T$:

$S \oplus T$
$S + T$
$S \mathop \triangle T$ or $S \mathop \Delta T$

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