Definition:Restriction/Relation
Definition
Let $\RR$ be a relation on $S \times T$.
Let $X \subseteq S$, $Y \subseteq T$.
The restriction of $\RR$ to $X \times Y$ is the relation on $X \times Y$ defined as:
- $\RR {\restriction_{X \times Y} }: = R \cap \paren {X \times Y}$
where $R \subseteq S \times T$ is the subset of the Cartesian product of $S$ and $T$ which defines the relation $\RR$.
If $Y = T$, then we simply call this the restriction of $\RR$ to $X$, and denote it as $\RR {\restriction_X}$.
A different way of saying the same thing is:
- $\RR {\restriction_X} = \set {\tuple {x, y} \in R: x \in X}$
Note that the parenthesis is not necessary in the above, but it does make the meaning clearer.
Class Theory
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation in $V$.
Let $A \subseteq V$ be a class.
The restriction of $\RR$ to $A$ is the relation on $A \times A$ defined as:
- $\RR {\restriction_A}: = \RR \cap \paren {A \times A}$
Notation
The use of the symbol $\restriction$ is a recent innovation over the more commonly-encountered $\vert$.
Thus the notation $\RR \vert_{X \times Y}$ and $\struct {T, \circ \vert_T}$, etc. are currently more likely to be seen than $\RR {\restriction_{X \times Y} }$ and $\struct {T, \circ {\restriction_T} }$.
No doubt as the convention becomes more established, $\restriction$ will develop.
It is strongly arguable that $\restriction$, affectionately known as the harpoon, is preferable to $\vert$ as the latter is suffering from the potential ambiguity of overuse.
Some authors prefer not to subscript the subset, and render the notation as:
- $f \mathbin \restriction X = \set {\tuple {x, \map f x}: x \in X}$
but this is not recommended on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it has less clarity.
Also note that it is commonplace even to omit the $\restriction$ symbol altogether, and merely render as $\RR_{X \times Y}$ or $\struct {T, \circ_T}$, and so on.
Also known as
Some sources refer to $\RR {\restriction_X}$ as the relation induced on $X$ by $\RR$.
Also see
- Results about restrictions can be found here.
Technical Note
The $\LaTeX$ code for \(f {\restriction_{X \times Y} }: X \to Y\) is f {\restriction_{X \times Y} }: X \to Y
.
Note that because of the way MathJax renders the image, the restriction symbol and its subscript \restriction_T
need to be enclosed within braces { ... }
in order for the spacing to be correct.
The $\LaTeX$ code for \(s \mathrel {\RR {\restriction_{X \times Y} } } t\) is s \mathrel {\RR {\restriction_{X \times Y} } } t
.
The $\LaTeX$ code for \(t_1 \mathbin {\circ {\restriction_T} } t_2\) is t_1 \mathbin {\circ {\restriction_T} } t_2
.
Again, note the use of \mathrel { ... }
and \mathbin { ... }
so as to render the spacing evenly.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.6$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.2$