Functions
section function_toolkit parents relation_toolkit |
Partial functions
generic 5 rightassoc (_ _) |
X Y == { f : X Y | p , q : f | p.1 = q.1 p.2 = q.2 } |
X
Y is the set of all partial functions from X to Y,
that is, the set of all relations between X and Y such that
each x in X is related to at most one y in Y.
The terms ``function'' and ``partial function'' are synonymous.
Partial injections
generic 5 rightassoc (_ _) |
X Y == { f : X Y | p, q : f p.1 = q.1 p.2 = q.2 } |
X
Y is the set of partial injections from X to Y,
that is, the set of all relations between X and Y such that
each x in X is related to no more than one y in Y,
and each y in Y is related to no more than one x in X.
The terms ``injection'' and ``partial injection'' are synonymous.
Total injections
generic 5 rightassoc (_ _) |
X
Y is the set of total injections from X to Y,
that is, the set of injections from X to Y
that are also total functions from X to Y.
Partial surjections
generic 5 rightassoc (_ _) |
X Y == { f : X Y | ran f = Y } |
X
Y is the set of partial surjections from X to Y,
that is, the set of functions from X to Y
whose range is equal to Y.
The terms ``surjection'' and ``partial surjection'' are synonymous.
Total surjections
generic 5 rightassoc (_ _) |
X
Y is the set of total surjections from X to Y,
that is, the set of surjections from X to Y
that are also total functions from X to Y.
Bijections
generic 5 rightassoc (_ _) |
X
Y is the set of bijections from X to Y,
that is, the set of total surjections from X to Y
that are also total injections from X to Y.
Finite functions
generic 5 rightassoc (_ _) |
The finite functions from X to Y are the functions from X to Y
that are also finite sets.
Finite injections
generic 5 rightassoc (_ _) |
The finite injections from X to Y are the injections from X to Y
that are also finite functions from X to Y.
Disjointness
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[L,X] |
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| disjoint _ : ( L X ) |
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| f : L X  |
| disjoint f ( p , q : f | p q p.2 q.2 = ) |
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A labelled family of sets is disjoint precisely when any
distinct pair yields sets with no members in common.
Partitions
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[L,X] |
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| _ partition _ : ( L X ) X |
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| f : L X ; a : X  |
| f partition a disjoint f ( ran f ) = a |
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A labelled family of sets f partitions a set a
precisely when f is disjoint and the union of all the sets in f
is a.
IT 5-Jan-2002