Discrete Mathematics. Chapter 4. Relations and Digraphs Sanguk Noh

Transcription

1 Discrete Mathematics Chapter 4. Relations and Digraphs Sanguk Noh

2 Table Product sets and partitions Relations and digraphs Paths in relations and digraphs Properties of relations Equivalence relations Operations on relations

3 Product sets and partitions Def.) product set or Cartesian product A B A B={(a,b) a A and b B} e.g.) A={,2,3} and B={r,s} A B={(,r),(,s),(2,r),(2,s),(3,r),(3,s)} Def.) partition or quotient set P of a nonempty set A Each element of A belongs to one of the sets in P. If A and A 2 are distinct elements of P, then A A 2 =. e.g.) A={a, b, c, d, e, f, g, h} A ={a,b,c,d}, A 2 ={a,c,e,f,g,h}, A 3 ={a,c,e,g}, A 4 ={b,d} A 5 ={f,h} {A, A 2 } : not a partition A A 2 P={A 3,A 4,A 5 } is a partition of A.

4 Relations and digraphs Def.) Let A and B be nonempty sets. Relation R from A to B A B. If R A B and (a,b) R, then a is related to b by R, a R b. When R A A, R is a relation on A. e.g.) A: the set of positive integers a R b iff a divides b, a b. Then, 4R2, 5R7. Def.) Let R A B be a relation from A to B. the domain of R, Dom(R) : the set of elements in A the range of R, Ran(R) : the set of elements in B the R-relative set of x, R(x)={y B x R y}, if R is a relation from A to B and x A. e.g.) A={a,b,c,d} and R A A. R={(a,a),(a,b),(b,c),(c,a),(d,c),(c,b)} If A ={c,d}, then R(A )={a,b,c}

5 Relations and digraphs Theorem : Let R be a relation from A to B, and let A A and A 2 A. If A A 2, then R(A ) R(A 2 ). R(A A 2 )=R(A ) R(A 2 ). R(A A 2 ) R(A ) R(A 2 ) e.g.) A={x x is an integer} R: A ={,,2}, A 2 ={9,3} R(A )={,,2, }, if x y. R(A 2 )={9,,, }, if x y. R(A ) R(A 2 )={9,,, } A A 2 =, R(A A 2 ) = n or n or 2 n

6 Relations and digraphs e.g.) Let A= {,2,3} and B={x,y,z,w,p,q}, and let R A B. Then, consider R={(,x),(,z),(2,w),(2,p),(2,q),(3,y)}. Let A ={,2} and A 2 ={2,3}. () R(A )={x,z,w,p,q} R(A 2 )={w,p,q,y} R(A ) R(A 2 )={x,y,z,w,p,q}=b Since A A 2 =A, R(A A 2 )=R(A)=B (2) R(A ) R(A 2 )={w,p,q}=r(a A 2 ) R(A ) R(A 2 ) R(A A 2 )

7 Relations and digraphs Def.) If A={a,a 2 a m }and B={b,b 2 b n } are finite sets, and R is a relation from A to B, then R: m n matrix M R =[m ij ], m ij = if (a i,b j ) R if (a i,b j ) R M R : the matrix of R. e.g.) A={,2,3}, B={r,s}, R={(,r),(2,s),(3,r)} the matrix of R, m n, M R = 2 3

8 Relations and digraphs Def.) digraph (or directed graph) of R vertices 2 3 Edge: a 3 R a e.g.) A={,2}, R={(,),(,2),(2,),(2,2)} R is a relation on A. 2

9 Relations and digraphs Def.) in-degree of a: the no. of b A s.t. (b,a) R out-degree of a: the no. of b A s.t. (a,b) R e.g.) The vertex in the previous figure has in-degree 2. e.g.) A={,4,5} R 4 5 Sol.) M R = R={(,4),(,5),(4,),(4,4),(5,4),(5,5)}

10 Relations and digraphs Def.) the restriction of R to B : R (B B) if R is a relation on a set A and B A. e.g.) R (B B) A={a,b,c}, B={a,b} R={(a,a),(a,c),(b,c),(b,a),(c,c)} B B={(a,a),(a,b),(b,a),(b,b)} R (B B) = {(a,a),(b,a)}

11 Paths in relations and digraphs Def.) a path of length n in R from a to b: π: a, x,x 2,,x n-, b such that a finite sequence arx, x Rx 2,, x n- Rb e.g.) 2 n+ elements π :,2,3 a path of length 2 from vertex to vertex 3 π 2 :,2,5, π 3 :2,3

12 Paths in relations and digraphs Def.) Cycle : a path that begins and ends at the same vertex. x R n y: There is a path of length n from x to y in R x R y: There is some path in R from x to y. connectivity relation for R. e.g.) A={a,b,c,d,e}, R={(a,a),(a,b),(b,c),(c,e),(c,d),(d,e)} (a) R 2 (b) R (a) R 2 ={(a,a),(a,b),(a,c),(b,d),(b,e),(c,e)} a R 2 a a R 2 b a R 2 c b R 2 d b R 2 e c R 2 e (b) R ={(a,a),(a,b),(a,c),(a,d),(a,e),(b,c,),(b,d),(b,e),(c,d),(c,e),(d,e)} a b d c e

13 Paths in relations and digraphs Theorem R is a relation on A ={a,a 2, a n }. M R2 =M R M R Proof) M R =[m ij ] M R2 =[n ij ] By the definition of M R M R, the i, jth element of M R M R is l iff the row i of M R and the column j of M R have a in the same relative position, say k. m ik = and m kj = for some k, k n. By the definition of M R, this means that a i R a k and a k R a j. Thus, a i R 2 a j, so n ij = M R M R = M R 2

14 Paths in relations and digraphs Theorem For n 2 and R a relation on a finite set A, we have M R n = M R M R M R (n factors). Proof by induction Basis step Let n=2. M 2 R = M R M R Induction hypothesis n=k for some k 2, M k R = M R.. M R (k factors) Induction step n=k+. M k+ R =[x ij ], M k R =[y ij ], and M R =[m ij ] if x ij =, we must have a path of length k+ from a i to a j. a i a s a j k

15 Paths in relations and digraphs y is = and m sj =, so M R k M R has a in position i,j. Similarly, if M R k M R has a l in position i,j, then x ij =. M R k+ = M R k M R M R k+ = M R k M R = (M R.. M R ) M R (k+ factors) Thus, by the induction, M R n = M R M R M R (n factors) is true for all n 2.

16 Paths in relations and digraphs Page 4, #26 A={,2,3,4,5}, R: arb iff a<b (a) R 2 and R 3 R 2 = R 3 = Then, R=? (b) a R 2 b iff? (c) a R 3 b iff?

17 Properties of relations Def.) reflexive a relation R on a set A( R A A) is reflexive if (a,a) R for all a A. e.g.) A={,2,3}, R={(,),(2,2),(3,3)}:reflexive cf.) irreflexive if (a,a) R for all a A. e.g.) R={(,),(2,3)} Is the empty relation reflexive?

18 Properties of relations Def.) Symmetric A relation R A A is symmetric if whenever arb, then bra. Def.) asymmetric If whenever arb, then bra Def.) antisymmetric If whenever arb and bra, then a=b if whenever a b, arb or bra

19 Properties of relations e.g.) A: the set of integers. R={(a,b) A A a<b} Sol.) Symmetric if a<b, then b a, R is not symmetric. Asymmetric if a<b, then b a, R is asymmetric. Antisymmetric if a b, then either a b or b a, R is antisymmetric

20 Properties of relations e.g.) A={,2,3,4}, R={(,2),(2,2),(3,4),(4,)} R: not symmetric (,2) R but (2,) R not asymmetric (2,2) R antisymmetric! Let M R =[m ij ]. symmetric if m ij =, then m ji = if m ij =, then m ji = 2. asymmetric if m ij =, then m ji = m ii = for all i if R is asymmetric. 3. antisymmetric if i j, then m ij = or m ji =.

21 Properties of relations e.g.) M R = M R2 = M R3 = Reflexive Irreflexive M R M R2 M R3 Symmetric Asymmetric antisymmetric

22 Properties of relations Digraph and graph of symmetric relation Let A= {a,b,c} and R A A. a c a c b b Digraph of R graph of R R =

23 Properties of relations Def.) Transitive If whenever a R b and b R c, then a R c. e.g.) A: the set of integers. R : the realtion < a R b and b R c. a,b,c A a<b and b<c, then a<c, so a R c. Hence R is transitive. e.g.) A={,2,3,4} R={(,2),(,3),(4,2)} transitive! e.g.) A={,2,3} M R = R={(,),(,2),(,3),(2,3),(3,3)} Since (,2) and (2,3), then (,3). Since (,3) and (3,3), then (,3).

24 Properties of relations Theorem R AxA Reflexivity of R a R(a) for all a in A Symmetry of R a R(b) iff b R(a) Transitivity of R if b R(a) and c R(b), then c R(a)

25 Equivalence relations Def.) A relation R on a set A is an equivalence if it is reflexive, symmetric, and transitive. e.g.) Let A={,2,3,4} and R={(,),(,2),(2,),(2,2),(3,4),(4,3),(3,3),(4,4)} reflexive? symmetric? transitive?

26 Equivalence relations Theorem Let P be a partition of a set A. Define the relation R on A: a R b iff a and b are members of the same block. Then R is an equivalence relation on A. Proof Reflexive : If a A, then a R a. ( a is in the same block) Symmetric : If a R b, a and b are in the same block. So. b R a Transitive : If a R b and b R c, then a, b, and c must be in the same block of P. Thus, a R c. R is the equivalence relation determined by P.

27 Equivalence relations Theorem Let R be an equivalence relation on A, and let P be the collection of all distinct relative sets R(a) for a in A. Then P is a partition of A, and R is the equivalence relation determined by P. Proof (a) Every element of A belongs to some relative set. (b) If R(a) and R(b) are not identical, then R(a) R(b)=. (a) is true, since a R(a) by reflexivity of R. (b) If R(a) R(b), then R(a)=R(b). Then a R c and b R c Since R is symmetric, c R b, and a R b by transitivity. By Lemma, R(a)=R(b), Thus, P is a partition. By Theorem, P determines R.

28 Equivalence relations e.g.) Let A={,2,3,4} and R={(,),(,2),(2,),(2,2),(3,4),(4,3),(3,3),(4,4)} Find partition of A Sol.) R() = {,2}=R(2) R(3) = {3,4}=R(4) Given R, hence, P={{,2},{3,4}} Page 52, problem 2. Let A={a,b,c,d,e} and R AxA. M R = a b d d e a b c d e partition of A : A/R=?

29 Equivalence relations Page 5, #4. A={,2,3,4,5}, P={{,3,5},{2,4}} equivalence relation R?

30 Operations on relations Operations R,S A B. complementary relation : R 2. intersection : R S 3. union : R S 4. inverse : R - : relation from B to A R - B A e.g.) A={,2,3,4}, B={a,b,c} R={(,a), (,b), (2,b), (2,c), (3,b), (4,a)} S={(,b),(2,c),(3,b),(4,b)}

31 Operations on relations Sol.) A B = {(,a),.,(4,c)} (a) R=? (b) R S (c) R S (d) R -

32 Operations on relations Closures reflexive closure of R a d a d b c b c R (does not possess a reflexive property) The reflexive closure of R Symmetric closure of R If A={a,b,c,d} and R={(a,b),(b,c),(a,c),(c,d)}, then R - ={(b,a), (c,b),(c,a),(d,c)} the symmetric closure of R: R R - ={(a,b),(b,a),(b,c),(c,b),(a,c),(c,a),(c,d),(d,c)}

33 Operations on relations Transitive closure of R a d b c R is not transitive Method : Finding transitivity from R Method 2: computing R Method 3: Warshall s algorithm

34 Warshall s algorithm e.g.) A= {,2,3,4}, R={(,2),(2,3),(3,4),(2,)} Method (,2),(2,3) (,3). (,2),(2,) (,) Method 2 2 R ={(,),(,2),(,3),(,4),(2,) (3,4)} 4 3

35 Warshall s algorithm Method 3 Warshall s algorithm How to implement the transitive closure of R! Boolean matrix W k A={a,a 2,,a n }, R A A, k n W k =[t ij ] t ij = iff there is a path from a i to a j in R whose interior vertices come from the set {a,a 2,,a k }. W n : iff some path in R connects a i with a j. W n = M R (since any vertex must come from the set {a,a 2,,a n }) Suppose W k =[t ij ] and W k- =[s ij ] () s ij = or (2) s ik = and s kj = t ij = Subpath a k Subpath 2 a i a j

36 Warshall s algorithm Algorithm Step : Copy W k- into W k. W =M R. Step 2: List the locations p,p 2, in column k of W k-, where the entry is, and the locations q,q 2, in row k of W k-, where the entry is. Step 3: Put s in all the positions p i, q j of W k.

37 Warshall s algorithm e.g.) A={,2,3,4}, R={(,2),(2,3),(3,4),(2,)} W =M R = k 4(=n). k= 2-, -2 i k k j 2. k=2 i 2,2 j k=3 3 i 3,3 4 j 2 w = w 2 = w 3 = 4. k=4 4, 4-(None of s) 2 3 No new s are added. M R =W 4 =W 3