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Reference manual / Standard toolkit / section function_toolkit


Functions

section function_toolkit parents relation_toolkit

Partial functions

generic 5 rightassoc (_ \pfun _)

X \pfun Y == {  f : X \rel Y | \forall p , q : f | p.1 = q.1 @ p.2 = q.2  }

X \pfun 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 (_ \pinj _)

X \pinj Y == {  f : X \rel Y | \forall p, q : f @ p.1 = q.1 \iff p.2 = q.2  }

X \pinj 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 (_ \inj _)

X \inj Y == ( X \pinj Y ) \cap ( X \fun Y )

X \inj 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 (_ \psurj _)

X \psurj Y == {  f : X \pfun Y | ran f = Y  }

X \psurj 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 (_ \surj _)

X \surj Y == ( X \psurj Y) \cap ( X \fun Y )

X \surj 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 (_ \bij _)

X \bij Y == ( X \surj Y ) \cap ( X \inj Y )

X \bij 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 (_ \ffun _)

X \ffun Y == (X \pfun Y) \cap \finset (X \cross Y)

The finite functions from X to Y are the functions from X to Y that are also finite sets.

Finite injections

generic 5 rightassoc (_ \finj _)

X \finj Y == ( X \ffun Y ) \cap ( X \pinj Y )

The finite injections from X to Y are the injections from X to Y that are also finite functions from X to Y.

Disjointness

relation (disjoint _)

[L,X]

disjoint _ : \power ( L \rel \power X )
\where \where \where
\forall f : L \rel \power X @
disjoint f \iff ( \forall p , q : f | p \neq q @ p.2 \cap q.2 = \emptyset )

A labelled family of sets is disjoint precisely when any distinct pair yields sets with no members in common.

Partitions

relation (_ partition _)

[L,X]

_ partition _ : ( L \rel \power X ) \rel \power X
\where \where \where
\forall f : L \rel \power X ; a : \power X @
f partition a \iff disjoint f \land \bigcup ( ran f ) = a

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