Definition:Restriction
Definition
Restriction of a Relation
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:
- $\RR {\restriction_{X \times Y} }: = \RR \cap \paren {X \times Y}$
Note that the parenthesis is not necessary in the above, but it does make the meaning clearer.
If $Y = T$, then we simply call this the restriction of $\RR$ to $X$, and denote it as $\RR {\restriction_X}$.
Restriction of a Mapping
Let $f: S \to T$ be a mapping.
Let $X \subseteq S$.
Let $f \sqbrk X \subseteq Y \subseteq T$.
The restriction of $f$ to $X \times Y$ is the mapping $f {\restriction_{X \times Y} }: X \to Y$ defined as:
- $f {\restriction_{X \times Y} } = f \cap \paren {X \times Y}$
If $Y = T$, then we simply call this the restriction of $f$ to $X$, and denote it as $f {\restriction_X}$.
Restriction of an Operation
In the same way that a restriction is defined on a relation, it can be defined on a binary operation.
Let $\struct {S, \circ}$ be an algebraic structure.
Let $A, B \subseteq S$.
The restriction of $\circ$ to $A \times B$ is denoted $\circ {\restriction_{A \times B} }$, and is defined as:
- $\forall a \in A, b \in B: a \mathbin {\circ {\restriction_{A \times B} } } b = a \circ b$
The notation $\circ {\restriction_{A \times B} }$ is generally used only if it is necessary to emphasise that $\circ {\restriction_{A \times B} }$ is strictly different from $\circ$ (through having a different domain).
When no confusion is likely to result, $\circ$ is generally used for both.
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 see
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.