# Definition:Relative Complement

## Contents

## Definition

Let $S$ be a set, and let $T \subseteq S$, that is: let $T$ be a subset of $S$.

Then the set difference $S \setminus T$ can be written $\relcomp S T$, and is called the **relative complement of $T$ in $S$**, or the **complement of $T$ relative to $S$**.

Thus:

- $\relcomp S T = \set {x \in S : x \notin T}$

## Illustration by Venn Diagram

The relative complement $\relcomp S T$ of the set $T$ with respect to $S$ is illustrated in the following Venn diagram by the red area:

## Also known as

Some authors call this **the complement** and use the term **relative complement** for the set difference $S \setminus T$ when the stipulation $T \subseteq S$ is not required.

Others emphasize the connection with set difference by referring to the **relative complement** as a **proper difference**.

Thus, in this view, the **relative complement** is a specific case of a set difference.

Some sources do away with the word **relative**, and refer just to the **complement of $T$ in $S$**

Different notations for $\relcomp S T$ mainly consist of variants of the $\complement$:

- $\map {\mathcal C_S} T$
- $\map {\mathbf C_S} T$
- $\map {c_S} T$
- $\map {C_S} T$
- $\operatorname C_S \paren T$

or sometimes:

- $\map {T\,^c} S$
- $\map {T\,^\complement} S$

... and sometimes the brackets are omitted:

- $C_S T$

If the superset $S$ is implicit, then it can be omitted: $\complement \paren T$ etc. See the notation for set complement.

Some sources do not bother to introduce a specific notation for the **relative complement**, and instead just use the various notation for set difference:

- $S \setminus T$
- $S / T$
- $S - T$

## Examples

### Examples

Let $E$ denote the set of all even integers.

Then the relative complement of $E$ in $\Z$:

- $\complement_\Z \paren E$

is the set of all odd integers.

## Also see

- Results about
**relative complement**can be found here.

## Linguistic Note

The word **complement** comes from the idea of **complete-ment**, it being the thing needed to **complete** something else.

It is a common mistake to confuse the words **complement** and **compliment**. Usually the latter is mistakenly used when the former is meant.

## Technical Note

The $\LaTeX$ code for \(\relcomp {S} {T}\) is `\relcomp {S} {T}`

.

This is a custom construct which has been set up specifically for the convenience of the users of $\mathsf{Pr} \infty \mathsf{fWiki}$.

Note that there are two arguments to this operator: the subscript, and the part between the brackets.

If either part is a single symbol, then the braces can be omitted, for example:

`\relcomp S T`

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 5$: Complements and Powers - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.6$. Difference and complement - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 3$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Sets - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Sets - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 6$: Subsets - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.2$: Outcomes and events: Footnote - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.2$: Boolean Operations - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 2$ - 2011: Robert G. Bartle and Donald R. Sherbert:
*Introduction to Real Analysis*(4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions