1. CS 103 Discrete Structures Lecture 19 Relations

2. Chapter 9

3. Chapter Summary Relations and Their Properties

4. Section 9.1

5. Section Summary Relations and Functions Properties of Relations Reflexive Relations Symmetric and Antisymmetric Relations Transitive Relations Combining Relations

6. Binary Relations A relation is a subset of the Cartesian product Relations can be used to solve problems such as: Determining which pairs of cities are linked by airline flights in a network Finding a feasible order for the different phases of a complicated project Producing a useful way to store information in computer databases A binary relation R from a set A to a set B is a subset R ⊆ A × B. Therefore, R consists of ordered pairs where the 1 st element of each ordered pair comes from A and the 2 nd element comes from B

7. Binary Relations (a, b)  R means a is related to b by R a R b denotes (a, b)  R a R b denotes (a, b)  R Example 1: Let A be the set of students, B be the set of courses and R be the relation that consists of pairs (a, b), where a is a student enrolled in course b. Then, If Ahmad and Ali are enrolled in CS103, then (Ahmad, CS103)  R and (Ali, CS103)  R If Ahmad is also enrolled in CS111, then (Ahmad, CS111)  R If Ali is not enrolled in CS111, then (Ali, CS111)  R

8. Binary Relations Example 2: Let A be the set of cities, B be the set of regions. (a, b) belongs to R if city a is in region b Then (Al-Mahd, Al-Madinah), (Gada, Makkah), and (Al-Zolfy, Ryiadh) are in R Example 3 A = {0, 1, 2} B = {a, b} R = {(0, a), (0, b), (1, a), (2, b)} is a relation from A to B We also can represent the relation from the set A to the set B graphically or using a table

9. Binary Relations on a Set A binary relation on the set A is a relation from A to A. In other words, a relation on a set A is a subset of A  A Example 1: A = {a, b, c}. Then R = {(a, a),(a, b), (a, c)} is a relation on A Example 2: A = {1, 2, 3, 4}. Which ordered pairs are in the relation R = {(a, b) | a divides b}? R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}

10. How Many Relations on a Set? Because a relation on A is the same thing as a subset of A × A, we count the subsets of A × A As A × A has |A| 2 elements, and a set with m elements has 2 m subsets, therefore A × A has 2 |A| 2 subsets  there are 2 |A| 2 relations on a set A

11. Binary Relations on a Set: Example 3 Which of these relations on the set of integers contain each of the pairs (1, 1), (1, 2), (2, 1), (1, -1), and (2, 2)? R 1 = {(a, b) | a ≤ b} R 2 = {(a, b) | a > b} R 3 = {(a, b) | a = b or a = - b} R 4 = {(a, b) | a = b} R 5 = {(a, b) | a = b + 1} R 6 = {(a, b) | a + b ≤ 3} (1, 1)is inR 1, R 3, R 4, and R 6 (1, 2)is inR 1 and R 6 (2, 1) is inR 2, R 5, and R 6 (1, -1)is inR 2, R 3, and R 6 (2, 2)is inR 1, R 3, and R 4 Note that these relations are on an infinite set and each of these relations is an infinite set

12. Properties of Relations: Reflexive Properties are used to classify relations on a set A relation R on a set A is called reflexive iff ∀ x[x ∊ A ⟶ (x, x) ∊ R], i.e. (a, a)  R for every element a  A A= {1, 2, 3, 4} R 1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)} R 2 = {(1, 1), (1, 2), (2, 1)} R 3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)} R 4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)} R 5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)} R 6 = {(3, 4)}

13. Properties of Relations: Irreflexive Relation R on a set A is irreflexive if (a, a)  R for all a  A Some relations are neither reflexive nor irreflexive Example Let A = {1, 2} and R = {(1, 1)} It is not reflexive, because (2, 2)  R It is not irreflexive, because (1, 1)  R

14. Properties of Relations: Symmetric Relation R on a set A is symmetric iff (b, a)  R whenever (a, b)  R, for all a, b  A Relation R on a set A is antisymmetric iff (a, b)  R & (b, a)  R then a = b, for all a, b  A Relations can be symmetric & antisymmetric, simultaneously A = {1, 2, 3, 4} R 1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)} R 2 = {(1, 1), (1, 2), (2, 1)} R 3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)} R 4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)} R 5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)} R 6 = {(3, 4)}

15. Antisymmetric Property: Example Details R 1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)} Whenever 2 nd -last column shows Yes, last column does not show Yes.  R 5 is not Antisymmetric (a, b)(a, b)(b, a)(b, a) (b, a)  R 4 ? a=b?a=b? (1, 1) Yes (1, 2)(1, 2)(2, 1)YesNo (2, 1)(1, 2)YesNo (2, 2) Yes (3, 4)(4, 3)No- (4, 1)(1, 4)No- (4, 4) Yes

16. Antisymmetric Property: Example Details R 4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)} Whenever 2 nd -last column shows Yes, last column also shows Yes.  R 5 is Antisymmetric (a, b)(a, b)(b, a)(b, a) (b, a)  R 4 ? a=b?a=b? (2, 1)(1, 2)No- (3, 1)(3, 1)(1, 3)(1, 3) - (3, 2)(2, 3)No- (4, 1)(1, 4)No- (4, 2)(2, 4)No- (4, 3)(3, 4)No-

17. Antisymmetric Property: Example Details R 5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)} Whenever 2 nd -last column shows Yes, last column also shows Yes.  R 5 is Antisymmetric (a, b)(a, b)(b, a)(b, a) (b, a)  R 4 ? a=b?a=b? (1, 1) Yes (1, 2)(1, 2)(2, 1)(2, 1)No- (1, 3)(3, 1)No- (1, 4)(4, 1)No- (2, 2) Yes (2, 3)(3, 2)No- (2, 4)(4, 2)No- (3, 3) Yes (3, 4)(4, 3)No- (4, 4) Yes

18. Properties of Relations: Transitive Relation R on a set A is transitive if whenever (a, b)  R and (b, c)  R, then (a, c)  R, for all a, b, c  A A = {1, 2, 3, 4} R 1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)} R 2 = {(1, 1), (1, 2), (2, 1)} R 3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)} R 4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)} R 5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)} R 6 = {(3, 4)}

19. Transitive Property: Example Details R 1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)} Start with the first pair, (1,1). Here a=1, b=1. Are their pairs that have their 1 st elements as b? Yes Only one pair, (1, 2). Here b=1, c=2 Is (a, c) a member of R 1 ? Yes.  R 1 can be transitive. Now move to the next pair, (1, 2). Here a=1, b=2. Are their pairs that have their 1 st elements as b? Yes Two pairs, (2, 1), (2, 2). Here b=2, c=1 and c=2 Is (a, c) a member of R 1 ? Yes for both c=1 and c=2.  R 1 may be transitive. Now move to the next pair, (2, 1). Here a=2, b=1. Are their pairs that have their 1 st elements as b? Yes Two pairs, (1, 1), (1, 2). Here b=1, c=1 and c=2 Is (a, c) a member of R 1 ? Yes for both c=1 and c=2.  R 1 may be transitive. Now move to the next pair, (2, 2). Here a=2, b=2. Are their pairs that have their 1 st elements as b? Yes Two pairs, (2, 1), (2, 2). Here b=2, c=1 and c=2 Is (a, c) a member of R 1 ? Yes for both c=1 and c=2.  R 1 may be transitive. Now move to the next pair, (3, 4). Here a=3, b=4. Are their pairs that have their 1 st elements as b? Yes Two pairs, (4, 1), (4, 4). Here b=4, c=1 and c=4 Is (a, c) a member of R 1 ? No for c=1.  R 1 is not transitive. Finished! No need to do any further checking.

20. Transitive Property: Example Details R 4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)} There are no “No” answers in the last column. Therefore, the relation R 4 is transitive. (a, b)(a, b) Pairs with b as their 1 st element c (a, c)  R 4 ? (2, 1)--- (3, 1)(3, 1)--- (3, 2)(2, 1)1Yes (4, 1)--- (4, 2)(2, 1)1Yes (4, 3)(3, 1) (3, 2) 1212 Yes

21. Properties of Relations: Summary Reflexive  a  A,(a, a)  R Irreflexive  a  A,(a, a)  R Symmetric  a,b  A,(a, b)  R  (b, a)  R Antisymmetric  a,b  A,(a, b)  R  (b, a)  R  a = b Transitive  a,b,c  A,(a, b)  R  (b, c)  R  (a, c)  R

22. Combining Relations Relations are sets They can be combined in any way sets can be combined Example 1 Let A = {1, 2, 3}, B = {1, 2, 3, 4}, R 1 = {(1, 1), (2, 2), (3, 3)} R 2 = {(1, 1), (1, 2), (1, 3), (1, 4)}

23. Combining Relations: Example 2 R 1 = {(x, y) | x y}, where x,y  R R 1  R 2 = ? (x, y)  R 1  R 2 iff (x, y)  R 1 or (x, y)  R 2 (x, y)  R 1  R 2 iff x y x y implies that x  y  R 1  R 2 = {(x, y) | x  y} R 1  R 2 =  as (x, y) cannot belong to both R 1 & R 2 R 1 - R 2 = R 1 R 2 - R 1 = R 2 R 1  R 2 = R 1  R 2 - R 1  R 2 = {(x, y) | x  y} x > y x < y y -axis x-axis x = yx = y

24. R is a relation from set A to set B S is a relation from set B to set C The composite of R and S, S  R, is the relation consisting of ordered pairs (a, c), where a  A, c  C, and for which there exists an element b  B such that (a, b)  R and (b, c)  S Example R is the relation from A = {1, 2, 3} to B = {1, 2, 3, 4} R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)} S is the relation from B = {1, 2, 3, 4} to C = {0, 1, 2} S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)} S  R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)} b b c c a a a a b b c c Composite of Relations

25. Powers of a Relation For relation R on set A, powers R n, n = 1, 2, 3,..., are defined recursively by Example R = {(1, 1), (2, 1), (3, 2), (4, 3)} = R 1 R 2 = R  R = {(1, 1), (2, 1), (3, 1), (4, 2)} R 3 = R 2  R = {(1, 1), (2, 1), (3, 1), (4, 1)} R 4 = R 3  R = {(1, 1), (2, 1), (3, 1), (4, 1)} = R 3 R n = R 3 for n = 5, 6, 7,.... as well

26. Section 9.1: Exercises