# 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}`

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### 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`

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### 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`

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### 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`

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### 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`

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

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### 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`

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### 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`

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

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### 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`

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### 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`

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### Empty Set

- $\O$

The **empty set**:

- $\O = \set {}$

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

or `\varnothing`

or `\empty`

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### 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`

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### 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`

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### 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`

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### Set Complement

Symbols:Set Theory/Set Complement

### 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`

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### 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`

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### 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}`

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### Set Equivalence

- $S \sim T$

Let $S$ and $T$ be sets.

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

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`

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The $\LaTeX$ code for \(S \nsim T\) is `S \nsim T`

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### 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`

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- The $\LaTeX$ code for \(A \stackrel {f} {\longrightarrow} B\) is
`A \stackrel {f} {\longrightarrow} B`

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- The $\LaTeX$ code for \(f: a \mapsto b\) is
`f: a \mapsto b`

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- The $\LaTeX$ code for \(f: A \rightarrowtail B\) is
`f: A \rightarrowtail B`

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- The $\LaTeX$ code for \(f: A \hookrightarrow B\) is
`f: A \hookrightarrow B`

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- The $\LaTeX$ code for \(f: A \twoheadrightarrow B\) is
`f: A \twoheadrightarrow B`

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- The $\LaTeX$ code for \(f: A \leftrightarrow B\) is
`f: A \leftrightarrow B`

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- The $\LaTeX$ code for \(f: A \stackrel {\sim} {\longrightarrow} B\) is
`f: A \stackrel {\sim} {\longrightarrow} B`

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### 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`

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### 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}`

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### 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`

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### 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`

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### 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`

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### 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.