1. 2.4 Operations on Relations
1.Inverse relation
Definition 2.13: Let R be a relation from A to B. The inverse relation of R is a relation from B to A, we write R-1, defined by R-1= {(b,a)|(a,b)R}

2. Theorem 2.1：Let R,R1, and R2 be relation from A to B. Then
(2)(R1∪R2)-1=R1-1∪R2-1;
(3)(R1∩R2)-1=R1-1∩R2-1;
(4)(A×B)-1=B×A;
(5)-1=;
(7)(R1-R2)-1=R1-1-R2-1
(8)If R1R2 then R1-1R2-1

3. Theorem 2. 2：Let R be a relation on A
Theorem 2.2：Let R be a relation on A. Then R is symmetric if only if R=R-1.
Proof: (1)If R is symmetric, then R=R-1。
RR-1 and R-1R。
(2)If R=R-1, then R is symmetric
For any (a,b)R, (b,a)?R

4. 2.Composition
Definition 2.14: Let R1 be a relation from A to B, and R2 be a relation from B to C. The composition of R1 and R2, we write R2R1, is a relation from A to C, and is defined R2R1={(a,c)|there exist some bB so that (a,b)R1 and (b,c)R2, where aA and cC}.
(1)R1 is a relation from A to B, and R2 is a relation from B to C
(2)commutative law? 
R1={(a1,b1), (a2,b3), (a1,b2)}
R2={(b4,a1), (b4,c1), (b2,a2), (b3,c2)}

5. Associative law?
For R1 A×B, R2B×C, and R3C×D
R3(R2R1)=?(R3R2)R1
subset of A×D
For any (a,d)R3(R2R1), (a,d)?(R3R2)R1,
Similarity, (R3R2)R1R3(R2R1)
Theorem 2.3：Let R1 be a relation from A to B, R2 be a relation from B to C, R3 be a relation from C to D. Then R3(R2R1)=(R3R2)R1(Associative law)

6. Definition 2. 15: Let R be a relation on A, and nN
Definition 2.15: Let R be a relation on A, and nN. The relation Rn is defined as follows.
(1)R0 ={(a,a)|aA}), we write IA.
(2)Rn+1=RRn.
Theorem 2.4: Let R be a relation on A, and m,nN. Then
(1)RmRn=Rm+n
(2)(Rm)n=Rmn

7. A={a1,a2,,an},B={b1,b2,,bm}
R1 and R2 be relations from A to B.
MR1=(xij), MR2=(yij)
MR1∪R2=(xijyij)
MR1∩R2=(xijyij)
 
Example:A={2,3,4},B={1,3,5,7}
R1={(2,3),(2,5),(2,7),(3,5),(3,7),(4,5),(4,7)}
R2={(2,5),(3,3),(4,1),(4,7)}
Inverse relation R-1 of R : MR-1=MRT, MRT is the transpose of MR.

8. A={a1,a2,,an},B={b1,b2,,bm}, C={c1,c2,,cr},
R1 be a relations from A to B, MR1=(xij)mn, R2 be a relation from B to C, MR2=(yij)nr. The composition R2R1 of R1 and R2,

9. Example：R={(a,b),(b,a),(a,c)},is not symmetric
+ (c,a),R'={(a,b),(b,a),(a,c), (c,a)}，R' is symmetric.
Closure

10. 2.5 Closures of Relations 1.Introduction
Construct a new relation R‘, s.t. RR’,
particular property,
smallest relation
closure
Definition 2.17: Let R be a relation on a set A. R' is called the reflexive(symmetric, transitive) closure of R, we write r(R)(s(R),t(R) or R+), if there is a relation R' with reflexivity (symmetry, transitivity) containing R such that R' is a subset of every relation with reflexivity (symmetry, transitivity) containing R.

11. Condition:
1)R' is reflexivity(symmetry, transitivity)
2)RR'
3)For any reflexive(symmetric, transitive) relation R", If RR", then R'R"
Example：If R is symmetric, s(R)=?
If R is symmetric，then s(R)=R
Contrariwise, If s(R)=R，then R is symmetric
R is symmetric if only if s(R)=R
Theorem 2.5: Let R be a relation on a set A. Then
(1)R is reflexive if only if r(R)=R
(2)R is symmetric if only if s(R)=R
(3)R is transitive if only if t(R)=R

12. Theorem 2.6: Let R1 and R2 be relations on A, and R1R2. Then
(1)r(R1)r(R2)；
(2)s(R1)s(R2)；
(3)t(R1)t(R2)。
Proof: (3)R1R2t(R1)t(R2)
Because R1R2， R1t(R2)
t(R2) :transitivity

13. 2.Computing closures Example:Let A={1,2,3},R={(1,2),(1,3)}. Then
Theorem 2.7: Let R be a relation on a set A, and IA be identity(diagonal) relation. Then r(R)=R∪IA(IA={(a,a)|aA})
Proof：Let R'=R∪IA.
Definition of closure
(1)For any aA, (a,a)?R'.
(2) R?R'.
(3)Suppose that R'' is reflexive and RR''，R'?R''

14. Theorem 2.8：Let R be a relation on a set A. Then s(R)=R∪R-1.
Proof：Let R'=R∪R-1
Definition of closure
(1) R', symmetric?
(2) R?R'.
(3)Suppose that R'' is symmetric and RR''，R'?R'')

15. Example ：symmetric closure of “<” on the set of integers,is“≠”
<,>，
Let A is no empty set.
The reflexive closure of empty relation on A is the identity relation on A
The symmetric closure of empty relation on A, is an empty relation.

16. Theorem 2.9: Let R be a relation on A. Then
Theorem 2.10：Let A be a set with |A|=n, and let R be a relation on A. Then

17. Example：A={a,b,c,d},R={(a,b),(b,a), (b,c),(c,d)}, t(R)=?

18. Exercise: P135 32,34
P139 1,2,12, 21,22,
P155 2,8,12,22,24,26,27,28,29,30
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Equivalence Relation, P 136, i.e 4.5
Partial order relations,P ,i.e 6.1,6.2