CISC 1400 Discrete Structures

Transcription

1 CISC 1400 Discrete Structures Chapter 4 Relations CISC1400 Yanjun Li 1 1 Relation A relation is a connection between objects in one set and objects in another set (possibly the same set). age is a relation defined between two sets: 1 st set: the set of names of students in your class. 2 nd set: the set of natural numbers. CISC1400 Yanjun Li 2 1

2 Describe a Relation in English Three things: 1 st set: domain of the relation 2 nd set: codomain of the relation Connection or rule is that links the objects in the domain and objects in the codomain. Example Domain: The set of names of people in your family. Codomain: {Red, Black, Brown, Blonde} Rule: (x,y) is in the relation if x s hair is y CISC1400 Yanjun Li 3 Describe a Relation in English What elements (objects) are in the following relation? Domain: The Natural Numbers, N Codomain: The Natural Numbers, N Rule: (x,y) is in the relation if x is one more than y (x = y+1) (1,0) (2,1) (3,2) CISC1400 Yanjun Li 4 2

3 Using Graph to Visualize Relation A cloud where everything falls within nodes (solid small circle): objects or elements (people, numbers, ) Arcs: connecting two elements that are related Arrows: the direction of the relation x -> y is related to 2, 2 is related to 3. 3 is related to CISC1400 Yanjun Li 5 5 Cartesian Product (Cross Product) A B Create a new set consisting of all possible ordered pairs where the first element is from A, and second element is from B. A B {( x, y) : x A and y B} CISC1400 Yanjun Li 6 3

4 Cartesian Product Example Set A = {sky, water, ground} Set B = {blue, gray, brown} A x B = {(sky, blue), (sky, gray), (sky, brown), (water, blue), (water, gray), (water, brown), (ground, blue), (ground, gray), (ground, brown)} CISC1400 Yanjun Li 7 Describe a Relation as a Subset of the Cartesian Product of Two Sets The Cartesian product is the relation that contains all possible connections between objects of its domain and codomain. Some relation is just a subset of the Cartesian product of two sets. Winter = {(sky, gray), (water, blue), (ground, brown)} Winter (A X B) Any relation r between A and B has to be a subset r (A X B) CISC1400 Yanjun Li 8 4

5 Exercise List elements in the relation r defined as follows: Domain: The Natural Numbers, N Codomain: The Natural Numbers, N Rule: r = {(x,y) NxN: x+y is odd} Adding two odd numbers? Adding two even numbers? Adding a number to itself? Adding an even number and an odd number? CISC1400 Yanjun Li 9 Reflexive A relation is reflexive if it contains ordered pairs that have the same element in both positions, e.g., (x,x) for all objects x. Or If r is a relation with domain and codomain A, then we say r (A x A) is reflexive if and only if: (x A) ((x,x) r) Or A relation is reflexive if every object in the domain is related to itself. CISC1400 Yanjun Li 10 5

6 Graphically Describe Reflexive If R is reflexive, there is a loop on each node in its graph A relation is not reflexive if there is some object in the domain that is not related to itself R CISC1400 Yanjun Li 11 Reflexive Example If set A consists of all people, and the relation is Is the same age as Let s pick several persons and see what happens Tom is the same age as Tom Carol is the same age as Carol Sally is the same age as Sally It always work, i.e., for any person, he/she is the same age as himself/herself. Thus is the same age as is reflexive CISC1400 Yanjun Li

7 Try this mathematical one Domain and codomain are N R={(x, y) N x N: (x + y) is even} Is R reflexive? Is any number in N related to itself under R? Try a few numbers, 1, 2, 3, For any numbers in N? Yes, since a number added to itself is always even (since 2 will be a factor), so R is reflexive. CISC1400 Yanjun Li Irreflexive vs. Non-reflexive If a relation has absolutely no (x,x) elements, we say that relation is irreflexive. An irreflexive relation s graph has no self-loop If a relation does not have (x,x) element for every x (even though there may be some x for which it is present), we say that relation is non-reflexive. CISC1400 Yanjun Li 14 7

8 Exercise Domain and codomain are N R={(x, y) N x N: x is larger than y} Is R reflexive? Is any number in N related to itself under R? Try a few numbers, 1, 2, 3, For any numbers in N? No, since a number is not larger than itself, we say it is not related to itself. Actually, no natural number is larger than itself, i.e., NOT related to itself Irreflexive relation CISC1400 Yanjun Li 15 For all relations No object is related to itself Irreflexive Some object is related to itself, some object is not related to itself Not reflexive, not irreflexive Every object is related to itself Reflexive All relations A relation cannot be both reflexive and irreflexive. CISC1400 Yanjun Li

9 Exercises: reflexive? irreflexive? Each of following relations is defined on set {1,2,3,4,5,6}, R : smaller or equal to R d : divides evenly : e.g., 2 divides 6 evenly, but 4 doesn t divides 6 evenly. R a : adds up to 6, e.g., (3,3), (1,5) R={(1,2),(3,4), (1,1)} CISC1400 Yanjun Li Symmetric A relation is symmetric if when it contains any ordered pair (x,y), it also contains the same pair in the other order (y,x) where x y. Or If r is a relation with domain and codomain A, then we say r (A x A) is symmetric if and only if for x y: ((x,y) r) ((y,x) r) Or In the graphs of symmetric relations, arcs go both ways (with two arrows) CISC1400 Yanjun Li 18 9

10 Symmetric Example has the same hair color as relation among a set of people Pick any two people, say A and B If A has the same color hair as B, then of course B has the same color hair as A Thus it is symmetric is a friend of, is the same age of, goes to same college as CISC1400 Yanjun Li 19 Is the following Relation Symmetric? Domain: {1, 2, 3,4} Relation={(1, 2), (1, 3), (4, 4), (4, 5), (3, 1), (5,4), (2, 1)} Yes, it is symmetric since (1,2) and (2,1) (1,3) and (3,1) (4,5) and (5,4) CISC1400 Yanjun Li

11 Antisymmetric vs. Non-symmetric A relation never has (y,x) when it has (x,y), we say that relation is antisymmetric. In the graph, anti-symmetric relations do not have two-way arcs. When a relation does not have (y,x) for all (x,y) (even though there may be some (x,y) for which it is present), we say that relation is non-symmetric CISC1400 Yanjun Li 21 Exercise is older than relation among a set of people If Sally is older than Tom, then Tom is older than Sally must be false We found a case where (Sally, Tom) is in the relation, but (Tom, Sally) is not in the relation Therefore, this relation is not symmetric. Actually, for is older than relation, it never works both way It is an Anti-symmetric relation. CISC1400 Yanjun Li 22 11

12 Bizarre Middle Ground Some relations are symmetric Some relations are anti-symmetric Some relations are non-symmetric, i.e., Some ordered pairs can be reversed Some ordered pairs cannot be reversed Anti-symmetric Non-symmetric All relations Symmetric CISC1400 Yanjun Li Non-symmetric Example knows the birthday of defined among a set of people Some pair of people know each other s birthday, Tom know the birthday of Sally Sally know the birthday of Tom So we now know that it is not anti-symmetric. Some people know the birthday of the other, but not vice versa Tom knows the birthday of Bill Bill doesn t know the birthday of Tom So the relation is not symmetric. Therefore this relation is neither symmetric nor anti-symmetric, it is non-symmetric. CISC1400 Yanjun Li

13 Exercises: Symmetric? Anti-symmetric? For each of following relations defined on set {1,2,3,4,5,6} R={(1,2),(3,4), (1,1),(2,1),(4,3)} R={(1,2),(3,4),(1,1)} R={(1,2),(3,4),(1,1),(4,3)} R : smaller or equal to R d : divides evenly : e.g., 2 divides 6 evenly R a : adds up to 6, e.g., (3,3), (1,5) CISC1400 Yanjun Li Properties of Relation Reflexive is about single objects of A. Symmetric is about pairs of objects of A. Transitive is about three pairs of objects of A. CISC1400 Yanjun Li 26 13

14 a c Transitive b A relation is transitive if it contains ordered pairs that allow shortcuts if (a,b) and (b,c) are both in the relation, then (a,c) must also be in the relation. Or If r is a relation with domain and codomain A, then we say r (A x A) is transitive if and only if : ((x,y) r and (y,z) r ) ((x,z) r) CISC1400 Yanjun Li 27 Transitive Example is older than relation among the set of people Alice is older than bob. i.e., (Alice, Bob) Bob is older than Cathy. i.e., (Bob, Cathy) Is Alice older than Cathy? Yes! Alice for sure is older than Cathy. We know that for any three people, A, B and C, if A is older than B, B is older than C, then fore sure, A is older than C. This relation is a transitive relation. CISC1400 Yanjun Li 28 14

15 Not Transitive A relation R is not transitive if there exists three elements in the domain, a, b, and c, and a is related to b - (a,b), b is related to c (b,c), but a is not related to c. The definition of transitive can only be tested for if there is a (b,c). Strangely enough, if there are no (b,c) cases ever in the relation we are testing, then the relation is also considered transitive. Not transitive Transitive All relations CISC1400 Yanjun Li 29 Is this relation transitive? is taking the same class as relation on a set of students Suppose three students, Bob, Katie, and Alex, Bob is taking the same class as Katie Katie is taking the same class as Alex And now consider: Is Bob is taking the same class as Alex? Many cases: no Katie takes 1100 with Bob, and takes history with Alex, where Bob and Alex has no classes in common. Therefore this relation is not transitive CISC1400 Yanjun Li

16 Exercises: Transitive or not? R : smaller or equal to defined on set {1,2,3,4} For three numbers a, b, c from {1,2,3,4} Would knowing that a b, and b c, allows me to conclude that a c? Yes! It s transitive! Let s check it s graph CISC1400 Yanjun Li Exercises: Transitive or not? Are the following relations defined on set {1,2,3,4,5,6} transitive? R={(1,2),(3,4), (1,1),(2,1),(4,3)} R={(1,2),(3,4),(1,1)} R={(1,2),(2,4),(1,1),(4,3)} R : smaller or equal to R d : divides evenly : e.g., 2 divides 6 evenly R a : adds up to 6, e.g., (3,3), (1,5) CISC1400 Yanjun Li

17 What properties does it have? Domain: (1, 2, 3, 4, 5) Codomain: (1, 2, 3, 4, 5) Relation={(1, 1), (1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (5,5)} The Answer is: It is not reflexive, and it is not irrefleixve It is not symmetric, and it is antisymmetric It is not transitive CISC1400 Yanjun Li What properties does it have? Domain: (1, 2, 3, 4, 5) Codomain: (1, 2, 3, 4, 5) Relation={(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} The Answer is: It is reflexive It is anti-symmetric It is transitive CISC1400 Yanjun Li

18 Exercise For the following relations defined on A={1,2,3,4,5,6,7,8} R={(1,1),(1,2),(1,4),(1,8),(2,2),(2,4),( 2,8),(3,3),(3,6),(4,4),(4,8),(5,5),(6,6),( 7,7),(8,8)} CISC1400 Yanjun Li Try a few out Consider all positive numbers. Identify properties for the following relations R1: Is less than or equal to R2: Is 10 less than CISC1400 Yanjun Li

19 Try a few out Consider following relations defined on the set of all people. Identify their properties: R 1 : Is smarter than R 2 : Went to the same high school as R 3 : Is a cousin of CISC1400 Yanjun Li