:: Relations and Their Basic Properties
:: by Edmund Woronowicz
::
:: Received March 15, 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 XBOOLE_0, TARSKI, ZFMISC_1, SUBSET_1, RELAT_1, VALUED_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, SUBSET_1;
constructors TARSKI, SUBSET_1, ZFMISC_1, XTUPLE_0, XBOOLE_0;
registrations XBOOLE_0, ZFMISC_1, XTUPLE_0;
requirements SUBSET, BOOLE;
begin
reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
definition
let IT be set;
attr IT is Relation-like means
:: RELAT_1:def 1
x in IT implies ex y,z st x = [y,z];
end;
registration
cluster empty -> Relation-like for set;
end;
definition
mode Relation is Relation-like set;
end;
reserve P,P1,P2,Q,R,S for Relation;
registration
let R be Relation;
cluster -> Relation-like for Subset of R;
end;
scheme :: RELAT_1:sch 1
RelExistence{A,B() -> set, P[object,object]}:
ex R being Relation st for x,y holds
[x,y] in R iff x in A() & y in B() & P[x,y];
definition
let P,R;
redefine pred P = R means
:: RELAT_1:def 2
for a,b holds [a,b] in P iff [a,b] in R;
end;
registration
let P,X;
cluster P /\ X -> Relation-like;
cluster P \ X -> Relation-like;
end;
registration
let P,R;
cluster P \/ R -> Relation-like;
end;
registration
let R,S be Relation;
cluster R \+\ S -> Relation-like;
end;
registration
let a, b be object;
cluster {[a,b]} -> Relation-like;
end;
registration
let a, b be set;
cluster [:a,b:] -> Relation-like;
end;
registration
let a, b, c, d be object;
cluster {[a,b],[c,d]} -> Relation-like;
end;
definition
let P,A;
redefine pred P c= A means
:: RELAT_1:def 3
for a,b holds [a,b] in P implies [a,b] in A;
end;
:: DOMAIN OF RELATION
notation
let R be Relation;
synonym dom R for proj1 R;
end;
:: CODOMAIN OF RELATION
notation
let R be Relation;
synonym rng R for proj2 R;
end;
::$CT 6
theorem :: RELAT_1:7
R c= [:dom R, rng R:];
theorem :: RELAT_1:8
R /\ [:dom R, rng R:] = R;
theorem :: RELAT_1:9
dom {[x,y]} = {x} & rng {[x,y]} = {y};
theorem :: RELAT_1:10
dom {[a,b],[x,y]} = {a,x} & rng {[a,b],[x,y]} = {b,y};
theorem :: RELAT_1:11
P c= R implies dom P c= dom R & rng P c= rng R;
theorem :: RELAT_1:12
rng(P \/ R) = rng P \/ rng R;
theorem :: RELAT_1:13
rng(P /\ R) c= rng P /\ rng R;
theorem :: RELAT_1:14
rng P \ rng R c= rng(P \ R);
:: FIELD OF RELATION
definition
::$CD 2
let R;
func field R -> set equals
:: RELAT_1:def 6
dom R \/ rng R;
end;
theorem :: RELAT_1:15
[a,b] in R implies a in field R & b in field R;
theorem :: RELAT_1:16
P c= R implies field P c= field R;
theorem :: RELAT_1:17
field {[x,y]} = {x,y};
theorem :: RELAT_1:18
field(P \/ R) = field P \/ field R;
theorem :: RELAT_1:19
field(P /\ R) c= field P /\ field R;
:: CONVERSE RELATION
definition
let R;
func R~ -> Relation means
:: RELAT_1:def 7
[x,y] in it iff [y,x] in R;
involutiveness;
end;
theorem :: RELAT_1:20
rng R = dom(R~) & dom R = rng(R~);
theorem :: RELAT_1:21
field R = field(R~);
theorem :: RELAT_1:22
(P /\ R)~ = P~ /\ R~;
theorem :: RELAT_1:23
(P \/ R)~ = P~ \/ R~;
theorem :: RELAT_1:24
(P \ R)~ = P~ \ R~;
:: COMPOSITION OF RELATIONS
definition
let P,R be set;
func P(#)R -> Relation means
:: RELAT_1:def 8
[x,y] in it iff ex z st [x,z] in P & [z,y] in R;
end;
notation
let P,R;
synonym P*R for P(#)R;
end;
theorem :: RELAT_1:25
dom(P*R) c= dom P;
theorem :: RELAT_1:26
rng(P*R) c= rng R;
theorem :: RELAT_1:27
rng R c= dom P implies dom(R*P) = dom R;
theorem :: RELAT_1:28
dom P c= rng R implies rng(R*P) = rng P;
theorem :: RELAT_1:29
P c= R implies Q*P c= Q*R;
theorem :: RELAT_1:30
P c= Q implies P*R c= Q*R;
theorem :: RELAT_1:31
P c= R & Q c= S implies P*Q c= R*S;
theorem :: RELAT_1:32
P*(R \/ Q) = (P*R) \/ (P*Q);
theorem :: RELAT_1:33
P*(R /\ Q) c= (P*R) /\ (P*Q);
theorem :: RELAT_1:34
(P*R) \ (P*Q) c= P*(R \ Q);
theorem :: RELAT_1:35
(P*R)~ = R~*P~;
theorem :: RELAT_1:36
(P*R)*Q = P*(R*Q);
:: EMPTY RELATION
registration
cluster non empty for Relation;
end;
registration
let f be non empty Relation;
cluster dom f -> non empty;
cluster rng f -> non empty;
end;
theorem :: RELAT_1:37
(for x,y holds not [x,y] in R) implies R = {};
theorem :: RELAT_1:38
dom {} = {} & rng {} = {};
theorem :: RELAT_1:39
{}*R = {} & R*{} = {};
registration
let X be empty set;
cluster dom X -> empty;
cluster rng X -> empty;
let R;
cluster X*R -> empty;
cluster R*X -> empty;
end;
theorem :: RELAT_1:40
field {} = {};
theorem :: RELAT_1:41
dom R = {} or rng R = {} implies R = {};
theorem :: RELAT_1:42
dom R = {} iff rng R = {};
registration
let X be empty set;
cluster X~ -> empty;
end;
theorem :: RELAT_1:43
{}~ = {};
theorem :: RELAT_1:44
rng R misses dom P implies R*P = {};
definition
let R be Relation;
attr R is non-empty means
:: RELAT_1:def 9
not {} in rng R;
end;
registration
cluster non-empty for Relation;
end;
registration
let R be Relation, S be non-empty Relation;
cluster R*S -> non-empty;
end;
:: IDENTITY RELATION
definition
let X;
func id X -> Relation means
:: RELAT_1:def 10
[x,y] in it iff x in X & x = y;
end;
registration let X;
reduce dom id X to X;
reduce rng id X to X;
end;
theorem :: RELAT_1:45
dom id X = X & rng id X = X;
registration let X;
reduce (id X)~ to id X;
end;
theorem :: RELAT_1:46
(id X)~ = id X;
theorem :: RELAT_1:47
(for x st x in X holds [x,x] in R) implies id X c= R;
theorem :: RELAT_1:48
[x,y] in (id X)*R iff x in X & [x,y] in R;
theorem :: RELAT_1:49
[x,y] in R*id Y iff y in Y & [x,y] in R;
theorem :: RELAT_1:50
R*(id X) c= R & (id X)*R c= R;
theorem :: RELAT_1:51
dom R c= X implies (id X)*R = R;
theorem :: RELAT_1:52
(id dom R)*R = R;
theorem :: RELAT_1:53
rng R c= Y implies R*(id Y) = R;
theorem :: RELAT_1:54
R*(id rng R) = R;
theorem :: RELAT_1:55
id {} = {};
theorem :: RELAT_1:56
rng P2 c= X & P2*R = id dom P1 & R*P1 = id X implies P1 = P2;
definition
let R,X;
func R|X -> Relation means
:: RELAT_1:def 11
[x,y] in it iff x in X & [x,y] in R;
end;
registration
let R be Relation, X be empty set;
cluster R|X -> empty;
end;
theorem :: RELAT_1:57
x in dom(R|X) iff x in X & x in dom R;
theorem :: RELAT_1:58
dom(R|X) c= X;
theorem :: RELAT_1:59
R|X c= R;
theorem :: RELAT_1:60
dom(R|X) c= dom R;
theorem :: RELAT_1:61
dom(R|X) = dom R /\ X;
theorem :: RELAT_1:62
X c= dom R implies dom(R|X) = X;
theorem :: RELAT_1:63
(R|X)*P c= R*P;
theorem :: RELAT_1:64
P*(R|X) c= P*R;
theorem :: RELAT_1:65
R|X = (id X)*R;
theorem :: RELAT_1:66
R|X = {} iff dom R misses X;
theorem :: RELAT_1:67
R|X = R /\ [:X,rng R:];
theorem :: RELAT_1:68
dom R c= X implies R|X = R;
registration let R;
reduce R|dom R to R;
end;
theorem :: RELAT_1:69
R|dom R = R;
theorem :: RELAT_1:70
rng(R|X) c= rng R;
theorem :: RELAT_1:71
(R|X)|Y = R|(X /\ Y);
registration let R,X;
reduce R|X|X to R|X;
end;
theorem :: RELAT_1:72
(R|X)|X = R|X;
theorem :: RELAT_1:73
X c= Y implies (R|X)|Y = R|X;
theorem :: RELAT_1:74
Y c= X implies (R|X)|Y = R|Y;
theorem :: RELAT_1:75
X c= Y implies R|X c= R|Y;
theorem :: RELAT_1:76
P c= R implies P|X c= R|X;
theorem :: RELAT_1:77
P c= R & X c= Y implies P|X c= R|Y;
theorem :: RELAT_1:78
R|(X \/ Y) = (R|X) \/ (R|Y);
theorem :: RELAT_1:79
R|(X /\ Y) = (R|X) /\ (R|Y);
theorem :: RELAT_1:80
R|(X \ Y) = R|X \ R|Y;
registration
let R be empty Relation, X be set;
cluster R|X -> empty;
end;
theorem :: RELAT_1:81
R|{} = {};
theorem :: RELAT_1:82
{}|X = {};
theorem :: RELAT_1:83
(P*R)|X = (P|X)*R;
definition
let Y,R;
func Y|`R -> Relation means
:: RELAT_1:def 12
[x,y] in it iff y in Y & [x,y] in R;
end;
registration
let R be Relation, X be empty set;
cluster X|`R -> empty;
end;
theorem :: RELAT_1:84
y in rng(Y|`R) iff y in Y & y in rng R;
theorem :: RELAT_1:85
rng(Y|`R) c= Y;
theorem :: RELAT_1:86
Y|`R c= R;
theorem :: RELAT_1:87
rng(Y|`R) c= rng R;
theorem :: RELAT_1:88
rng(Y|`R) = rng R /\ Y;
theorem :: RELAT_1:89
Y c= rng R implies rng(Y|`R) = Y;
theorem :: RELAT_1:90
(Y|`R)*P c= R*P;
theorem :: RELAT_1:91
P*(Y|`R) c= P*R;
theorem :: RELAT_1:92
Y|`R = R*(id Y);
theorem :: RELAT_1:93
Y|`R = R /\ [:dom R,Y:];
theorem :: RELAT_1:94
rng R c= Y implies Y|`R = R;
registration let R;
reduce rng R|`R to R;
end;
theorem :: RELAT_1:95
rng R|`R=R;
theorem :: RELAT_1:96
Y|`(X|`R) = (Y /\ X)|`R;
registration let Y,R;
reduce Y|`(Y|`R) to Y|`R;
end;
theorem :: RELAT_1:97
Y|`(Y|`R) = Y|`R;
theorem :: RELAT_1:98
X c= Y implies Y|`(X|`R) = X|`R;
theorem :: RELAT_1:99
Y c= X implies Y|`(X|`R) = Y|`R;
theorem :: RELAT_1:100
X c= Y implies X|`R c= Y|`R;
theorem :: RELAT_1:101
P1 c= P2 implies Y|`P1 c= Y|`P2;
theorem :: RELAT_1:102
P1 c= P2 & Y1 c= Y2 implies Y1|`P1 c= Y2|`P2;
theorem :: RELAT_1:103
(X \/ Y)|`R = (X|`R) \/ (Y|`R);
theorem :: RELAT_1:104
(X /\ Y)|`R = X|`R /\ Y|`R;
theorem :: RELAT_1:105
(X \ Y)|`R = X|`R \ Y|`R;
theorem :: RELAT_1:106
{}|`R = {};
theorem :: RELAT_1:107
Y|`{} = {};
theorem :: RELAT_1:108
Y|`(P*R) = P*(Y|`R);
theorem :: RELAT_1:109
(Y|`R)|X = Y|`(R|X);
:: IMAGE OF SET IN RELATION
definition
let R,X;
func R.:X -> set means
:: RELAT_1:def 13
y in it iff ex x st [x,y] in R & x in X;
end;
theorem :: RELAT_1:110
y in R.:X iff ex x st x in dom R & [x,y] in R & x in X;
theorem :: RELAT_1:111
R.:X c= rng R;
theorem :: RELAT_1:112
R.:X = R.:(dom R /\ X);
theorem :: RELAT_1:113
R.:dom R = rng R;
theorem :: RELAT_1:114
R.:X c= R.:(dom R);
theorem :: RELAT_1:115
rng(R|X) = R.:X;
registration let R; let X be empty set;
cluster R.:X -> empty;
end;
registration let R be empty Relation; let X;
cluster R.:X -> empty;
end;
::$CT 2
theorem :: RELAT_1:118
R.:X = {} iff dom R misses X;
theorem :: RELAT_1:119
X <> {} & X c= dom R implies R.:X <> {};
theorem :: RELAT_1:120
R.:(X \/ Y) = R.:X \/ R.:Y;
theorem :: RELAT_1:121
R.:(X /\ Y) c= R.:X /\ R.:Y;
theorem :: RELAT_1:122
R.:X \ R.:Y c= R.:(X \ Y);
theorem :: RELAT_1:123
X c= Y implies R.:X c= R.:Y;
theorem :: RELAT_1:124
P c= R implies P.:X c= R.:X;
theorem :: RELAT_1:125
P c= R & X c= Y implies P.:X c= R.:Y;
theorem :: RELAT_1:126
(P*R).:X = R.:(P.:X);
theorem :: RELAT_1:127
rng(P*R) = R.:(rng P);
theorem :: RELAT_1:128
(R|X).:Y c= R.:Y;
theorem :: RELAT_1:129
for R be Relation, X, Y be set st X c= Y holds (R|Y).:X = R.:X;
theorem :: RELAT_1:130
(dom R) /\ X c= (R~).:(R.:X);
:: INVERSE IMAGE OF SET IN RELATION
definition
let R,Y;
func R"Y -> set means
:: RELAT_1:def 14
x in it iff ex y st [x,y] in R & y in Y;
end;
theorem :: RELAT_1:131
x in R"Y iff ex y st y in rng R & [x,y] in R & y in Y;
theorem :: RELAT_1:132
R"Y c= dom R;
theorem :: RELAT_1:133
R"Y = R"(rng R /\ Y);
theorem :: RELAT_1:134
R"rng R = dom R;
theorem :: RELAT_1:135
R"Y c= R"rng R;
registration let R; let Y be empty set;
cluster R"Y -> empty;
end;
registration let R be empty Relation; let Y;
cluster R"Y -> empty;
end;
::$CT 2
theorem :: RELAT_1:138
R"Y = {} iff rng R misses Y;
theorem :: RELAT_1:139
Y <> {} & Y c= rng R implies R"Y <> {};
theorem :: RELAT_1:140
R"(X \/ Y) = R"X \/ R"Y;
theorem :: RELAT_1:141
R"(X /\ Y) c= R"X /\ R"Y;
theorem :: RELAT_1:142
R"X \ R"Y c= R"(X \ Y);
theorem :: RELAT_1:143
X c= Y implies R"X c= R"Y;
theorem :: RELAT_1:144
P c= R implies P"Y c= R"Y;
theorem :: RELAT_1:145
P c= R & X c= Y implies P"X c= R"Y;
theorem :: RELAT_1:146
(P*R)"Y = P"(R"Y);
theorem :: RELAT_1:147
dom(P*R) = P"(dom R);
theorem :: RELAT_1:148
(rng R) /\ Y c= (R~)"(R"Y);
begin :: Addenda
definition
let R;
attr R is empty-yielding means
:: RELAT_1:def 15
rng R c= {{}};
end;
theorem :: RELAT_1:149
R is empty-yielding iff for X st X in rng R holds X = {};
:: from AMI_3
theorem :: RELAT_1:150
for f,g being Relation, A,B being set st f|A = g|A & f|B = g|B holds
f|(A \/ B) = g|(A \/ B);
theorem :: RELAT_1:151
for X being set, f,g being Relation st dom g c= X & g c= f holds g c= f|X;
theorem :: RELAT_1:152
for f being Relation, X being set st X misses dom f holds f|X = {};
:: from AMI_5
theorem :: RELAT_1:153
for f,g being Relation, A,B being set st A c= B & f|B = g|B holds f|A = g|A;
:: missing, 2005.07.11, A.T.
theorem :: RELAT_1:154
R|dom S = R|dom(S|dom R);
:: missing, 2005.11.13, A.T.
registration
cluster empty -> empty-yielding for Relation;
end;
registration
let R be empty-yielding Relation;
let X be set;
cluster R|X -> empty-yielding;
end;
theorem :: RELAT_1:155
R|X is non empty-yielding implies R is non empty-yielding;
:: from EQREL_1 (Class), 2007.02.06
definition
let R be Relation, x be object;
func Im(R,x) -> set equals
:: RELAT_1:def 16
R.:{x};
end;
:: from YELLOW_3, 2007.06.13, A.T.
scheme :: RELAT_1:sch 2
ExtensionalityR { A, B() -> Relation, P[object,object] }:
A() = B()
provided
for a, b being object holds [a,b] in A() iff P[a,b] and
for a, b being object holds [a,b] in B() iff P[a,b];
:: from SCMFSA6A, 2007.07.23, A.T.
theorem :: RELAT_1:156
dom (R | (dom R \ X)) = dom R \ X;
theorem :: RELAT_1:157
R | X = R | (dom R /\ X);
:: missing, 2007.11.07, A.T.
theorem :: RELAT_1:158
dom [:X,Y:] c= X;
theorem :: RELAT_1:159
rng [:X,Y:] c= Y;
:: from FUNCOP_1, 2008.02.19, A.T.
theorem :: RELAT_1:160
X <> {} & Y <> {} implies dom [:X,Y:] = X & rng [:X,Y:] = Y;
theorem :: RELAT_1:161
dom R = {} & dom Q = {} implies R = Q;
theorem :: RELAT_1:162
rng R = {} & rng Q = {} implies R = Q;
theorem :: RELAT_1:163
dom R = dom Q implies dom(S*R) = dom (S*Q);
theorem :: RELAT_1:164
rng R = rng Q implies rng(R*S) = rng (Q*S);
:: from RELSET_2 (R"x, generalized), 2008.07.16, A.T.
definition
let R be Relation, x be object;
func Coim(R,x) -> set equals
:: RELAT_1:def 17
R"{x};
end;
:: from NECKLACE, 2008.07.25, A.T.
registration
let R be trivial Relation;
cluster dom R -> trivial;
end;
registration
let R be trivial Relation;
cluster rng R -> trivial;
end;
:: comp. RFUNCT_2:34, CFCONT_1:31, 2008.08.07, A.T.
theorem :: RELAT_1:165
rng R c= dom (S|X) implies R*(S|X) = R*S;
:: from LATTICE2, 2008.09.14, A.T.
theorem :: RELAT_1:166
Q|A = R|A implies Q.:A = R.:A;
:: new, 2009.01.21, A.T
definition
let X,R;
attr R is X-defined means
:: RELAT_1:def 18
dom R c= X;
attr R is X-valued means
:: RELAT_1:def 19
rng R c= X;
end;
registration
let X,Y;
cluster X-defined Y-valued for Relation;
end;
:: from QC_LANG4, 2009.01.23, A.T
theorem :: RELAT_1:167
for D being set, R being D-valued Relation
for y being object st y in rng R holds y in D;
:: new, 2009.02.07, A.T.
registration
let X,A;
let R be A-valued Relation;
cluster R|X -> A-valued;
end;
registration
let X,A;
let R be A-defined Relation;
cluster R|X -> A-defined X-defined;
end;
registration
let X;
cluster id X -> X-defined X-valued;
end;
:: 2009.02.16, A.T.
registration
let A be set;
let R be A-valued Relation, S be Relation;
cluster S*R -> A-valued;
end;
registration
let A be set;
let R be A-defined Relation, S be Relation;
cluster R*S -> A-defined;
end;
:: 2009.02.16, A.T.
theorem :: RELAT_1:168
x in X implies Im([:X,Y:],x) = Y;
theorem :: RELAT_1:169
[x,y] in R iff y in Im(R,x);
theorem :: RELAT_1:170
x in dom R iff Im(R,x) <> {};
theorem :: RELAT_1:171
{} is X-defined Y-valued;
:: 2009.10.02, A.T.
registration
cluster empty -> non-empty for Relation;
end;
registration let X be set, R be X-defined Relation;
cluster -> X-defined for Subset of R;
end;
registration let X be set, R be X-valued Relation;
cluster -> X-valued for Subset of R;
end;
:: missing, 2010.01.02, A.T
theorem :: RELAT_1:172
X misses Y implies R|X|Y = {};
:: from AMISTD_3, 2010.01.10, A.T.
theorem :: RELAT_1:173
field {[x,x]} = {x};
registration let X; let R be X-defined Relation;
reduce R|X to R;
end;
registration let Y; let R be Y-valued Relation;
reduce Y|`R to R;
end;
theorem :: RELAT_1:174
for R being X-defined Relation holds R = R|X;
theorem :: RELAT_1:175
for S being Relation, R being X-defined Relation st R c= S holds R c= S|X;
:: missing, 2010.04.07, A.T.
theorem :: RELAT_1:176
dom R c= X implies R \ (R|A) = R|(X\A);
theorem :: RELAT_1:177
dom(R \ (R|A)) = dom R \ A;
theorem :: RELAT_1:178
dom R \ dom(R|A) = dom(R \ (R|A));
theorem :: RELAT_1:179
dom R misses dom S implies R misses S;
theorem :: RELAT_1:180
rng R misses rng S implies R misses S;
theorem :: RELAT_1:181
X c= Y implies (R \ R|Y)|X = {};
theorem :: RELAT_1:182
X c= Y implies
for R being X-defined Relation holds R is Y-defined;
theorem :: RELAT_1:183
X c= Y implies
for R being X-valued Relation holds R is Y-valued;
theorem :: RELAT_1:184
R c= S iff R c= S|dom R;
theorem :: RELAT_1:185
for R being X-defined Y-valued Relation holds R c= [:X,Y:];
:: new, 20011.06.11, A.T.
theorem :: RELAT_1:186
dom(X|`R) c= dom R;
registration
let A,X;
let R be A-defined Relation;
cluster X|`R -> A-defined;
end;
registration
let X,A;
let R be A-valued Relation;
cluster X|`R -> A-valued X-valued;
end;
registration
let X be empty set;
cluster -> empty for X-defined Relation;
cluster -> empty for X-valued Relation;
end;
:: from PNPROC_1, 2012.02.20, A.T.
theorem :: RELAT_1:187
for X,Y being set, P,R being Relation st X misses Y holds P|X misses R|Y;
theorem :: RELAT_1:188
for Y being set, R being Relation holds Y|`R c= R|(R"Y);
theorem :: RELAT_1:189
for f being Relation, x, y st dom f = {x} & rng f = {y}
holds f = { [x,y] };
registration
let X,Y;
sethood of X-defined Y-valued Relation;
end;