# Symbols:Set Operations and Relations

## Contents

- 1 Set Delimiters
- 2 Empty Set
- 3 Set Intersection
- 4 Set Union
- 5 Ordered Sum
- 6 Set Difference
- 7 Cartesian Product
- 8 Is an Element Of
- 9 Contains as an Element
- 10 Universal Quantifier
- 11 Existential Quantifier
- 12 Cardinality
- 13 Subset
- 14 Proper Subset
- 15 Superset
- 16 Proper Superset
- 17 Negation
- 18 Mappings
- 19 Alternative Symbols
- 20 Deprecated Symbols

## 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 Cartesian product.

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 $S$.

- $\forall x: P \paren x$ means that the propositional function $P \paren x$ is true
*for every*$x$ in the universal set.

The $\LaTeX$ code for \(\forall\) is `\forall`

.

## Existential Quantifier

- $\exists$

"There exists".

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

- $\exists x: P \paren x$ means that
*there exists at least one*$x$ in the universal set 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.