:: Relations Defined on Sets :: by Edmund Woronowicz :: :: Received April 14, 1989 :: Copyright (c) 1990-2018 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies RELAT_1, SUBSET_1, ZFMISC_1, TARSKI, XBOOLE_0; notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1; constructors TARSKI, SUBSET_1, RELAT_1, XTUPLE_0; registrations XBOOLE_0, RELAT_1, SUBSET_1; requirements SUBSET, BOOLE; begin reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object; :: :: RELATION AS A SUBSET OF CARTESIAN PRODUCT OF TWO SETS :: _____________________________________________________ definition let X,Y; mode Relation of X,Y is Subset of [:X,Y:]; end; registration let X,Y; cluster -> Relation-like for Subset of [:X,Y:]; end; registration let X,Y; cluster -> X-defined Y-valued for Relation of X,Y; end; reserve P,R for Relation of X,Y; definition let X,Y,R,Z; redefine pred R c= Z means :: RELSET_1:def 1 for x being Element of X, y being Element of Y holds [x,y] in R implies [x,y] in Z; end; definition let X,Y,P,R; redefine pred P = R means :: RELSET_1:def 2 for x being Element of X, y being Element of Y holds [x,y] in P iff [x,y] in R; end; theorem :: RELSET_1:1 A c= R implies A is Relation of X,Y; theorem :: RELSET_1:2 a in R implies ex x,y st a = [x,y] & x in X & y in Y; theorem :: RELSET_1:3 x in X & y in Y implies {[x,y]} is Relation of X,Y; theorem :: RELSET_1:4 for R being Relation st dom R c= X & rng R c= Y holds R is Relation of X,Y; theorem :: RELSET_1:5 dom R c= X1 implies R is Relation of X1,Y; theorem :: RELSET_1:6 rng R c= Y1 implies R is Relation of X,Y1; theorem :: RELSET_1:7 X c= X1 & Y c= Y1 implies R is Relation of X1,Y1; registration let X; let R,S be X-defined Relation; cluster R \/ S -> X-defined; end; registration let X; let R be X-defined Relation, S be Relation; cluster R /\ S -> X-defined; cluster R \ S -> X-defined; end; registration let X; let R,S be X-valued Relation; cluster R \/ S -> X-valued; end; registration let X; let R be X-valued Relation, S be Relation; cluster R /\ S -> X-valued; cluster R \ S -> X-valued; end; definition let X; let R be X-defined Relation; redefine func dom R -> Subset of X; end; definition let X; let R be X-valued Relation; redefine func rng R -> Subset of X; end; theorem :: RELSET_1:8 field R c= X \/ Y; theorem :: RELSET_1:9 (for x being object st x in X ex y being object st [x,y] in R) iff dom R = X; theorem :: RELSET_1:10 (for y being object st y in Y ex x being object st [x,y] in R) iff rng R = Y; definition let X,Y,R; redefine func R~ -> Relation of Y,X; end; definition let X,Y1,Y2,Z; let P be Relation of X,Y1; let R be Relation of Y2,Z; redefine func P*R -> Relation of X,Z; end; theorem :: RELSET_1:11 dom (R~) = rng R & rng (R~) = dom R; theorem :: RELSET_1:12 {} is Relation of X,Y; registration let A be empty set, B be set; cluster -> empty for Relation of A,B; cluster -> empty for Relation of B,A; end; theorem :: RELSET_1:13 id X c= [:X,X:]; theorem :: RELSET_1:14 id X is Relation of X,X; theorem :: RELSET_1:15 id A c= R implies A c= dom R & A c= rng R; theorem :: RELSET_1:16 id X c= R implies X = dom R & X c= rng R; theorem :: RELSET_1:17 id Y c= R implies Y c= dom R & Y = rng R; definition let X,Y,R,A; redefine func R|A -> Relation of X,Y; end; definition let X,Y,B,R; redefine func B|`R -> Relation of X,Y; end; theorem :: RELSET_1:18 R|X1 is Relation of X1,Y; theorem :: RELSET_1:19 X c= X1 implies R|X1 = R; theorem :: RELSET_1:20 Y1|`R is Relation of X,Y1; theorem :: RELSET_1:21 Y c= Y1 implies Y1|`R = R; definition let X,Y,R,A; redefine func R.:A -> Subset of Y; redefine func R"A -> Subset of X; end; theorem :: RELSET_1:22 R.:X = rng R & R"Y = dom R; theorem :: RELSET_1:23 R.:(R"Y) = rng R & R"(R.:X) = dom R; scheme :: RELSET_1:sch 1 RelOnSetEx{A() -> set,B() -> set,P[object,object]}: ex R being Relation of A(),B() st for x,y holds [x,y] in R iff x in A() & y in B() & P[x,y]; :: Relation on a set definition let X; mode Relation of X is Relation of X,X; end; reserve D,D1,D2,E,F for non empty set; reserve R for Relation of D,E; reserve x for Element of D; reserve y for Element of E; registration let D be non empty set; cluster id D -> non empty; end; theorem :: RELSET_1:24 for x being Element of D holds x in dom R iff ex y being Element of E st [x,y] in R; theorem :: RELSET_1:25 for y being object holds y in rng R iff ex x being Element of D st [x,y] in R; theorem :: RELSET_1:26 dom R <> {} implies ex y being Element of E st y in rng R; theorem :: RELSET_1:27 rng R <> {} implies ex x being Element of D st x in dom R; theorem :: RELSET_1:28 for P being Relation of D,E, R being Relation of E,F for x, z being object holds [x,z] in P*R iff ex y being Element of E st [x,y] in P & [y,z] in R; theorem :: RELSET_1:29 y in R.:D1 iff ex x being Element of D st [x,y] in R & x in D1; theorem :: RELSET_1:30 x in R"D2 iff ex y being Element of E st [x,y] in R & y in D2; scheme :: RELSET_1:sch 2 RelOnDomEx{A,B() -> non empty set, P[object,object]}: ex R being Relation of A(),B () st for x being Element of A(), y being Element of B() holds [x,y] in R iff P [x,y]; begin :: Addenda :: missing, 2006.11.04, A.T. scheme :: RELSET_1:sch 3 { N()-> set, M() -> Subset of N(), F(object)->set }: ex R being Relation of M() st for i being Element of N() st i in M() holds Im(R,i) = F(i) provided for i being Element of N() st i in M() holds F(i) c= M(); theorem :: RELSET_1:31 for N being set, R,S being Relation of N st for i being set st i in N holds Im(R,i) = Im(S,i) holds R = S; :: from AMI_3, 2007.06.13, A.T. (generalized) scheme :: RELSET_1:sch 4 {A,B() -> set, P[set,set], P,R()->Relation of A(), B()}: P() = R() provided for p being Element of A(), q being Element of B() holds [p,q] in P( ) iff P[p,q] and for p being Element of A(), q being Element of B() holds [p,q] in R( ) iff P[p,q]; :: missing, 2007.06.14, A.T. registration let X,Y,Z; let f be Relation of [:X,Y:], Z; cluster dom f -> Relation-like; end; registration let X,Y,Z; let f be Relation of X, [:Y, Z:]; cluster rng f -> Relation-like; end; :: missing, 2008.04.30, A.T. theorem :: RELSET_1:32 A misses X implies P|A = {}; :: missing, 2008.09.15, A.T. registration let R be non empty Relation, Y be non empty Subset of dom R; cluster R|Y -> non empty; end; registration let R be non empty Relation; let Y be non empty Subset of dom R; cluster R.:Y -> non empty; end; :: missing, 2008.11.29 A.T. registration let X,Y be set; cluster empty for Relation of X,Y; end; scheme :: RELSET_1:sch 5 {A() -> set,B() -> set,P[object,object]}: ex R being Relation of A(),B() st for x,y being set holds [x,y] in R iff x in A() & y in B() & P[x,y];