binary relation:

1. Discrete Math for CS Binary Relation: A binary relation between sets A and B is a subset of the Cartesian Product A x B. If A = B we say that the relation is a relation on A. Suppose A = {1, 3, 5, 7} and B = {2, 4, 6, 8}. Further suppose R = { (1,2), (3,4), (5,6), (7,8)}. This is a subset of A x B so is a binary relation between A and B

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3. Discrete Math for CS Binary Relations as Sets of Ordered Pairs: Because we mention one set before another in a Cartesian Product, A x B, the element, (a,b), in any relation, R, over A and B must have its first element from A and its second element from B. So we say that the elements of R form ordered pairs.

4. Discrete Math for CS Exercise: If X = {1, 2, 3, 4, 5, 6}, find R = { (x,y): x is a divisor of y}

5. Discrete Math for CS Graph Representation of a Binary Relation: If A and B are two finite sets and R is a binary relation between A and B we can represent this relation as a graph (set of vertices and edges).

6. Discrete Math for CS Example: A = {1, 2, 3, 4, 5, 6}. R is a relation on A defined by the following directed graph.

7. Discrete Math for CS Matrix Representation of a Binary Relation: If A and B are finite sets and R is a binary relation between A and B then create a matrix, M, with the following properties: the rows of the matrix are indexed by the elements of A the columns of the matrix are indexed by elements of B M(ai,bj) = 1 if (ai,bj) belongs to R; 0 otherwise

8. Discrete Math for CS Notation: If R is a binary relation on a set, X, we write x R y whenever (x,y) e R.

9. Discrete Math for CS RelationProperties: Suppose R is a relation on a set A. We say R is reflexive if a R a for all a e A. R is symmetric if x R y ==> y R x for all x, y e A. R is antisymmetric when (x R y and y R x ==> x == y) for all x, y e A. R is transitive when (x R y and y R z ==> x R z) for all x, y, z e A.

10. Discrete Math for CS Understanding Relations as Ordered Pairs: R is reflexive if (x,x) e R for all x e A. R is symmetric if when (x,y) e R then (y,x) e R for all x,y e A. R is antisymmetric if when (x,y) e R and x != y then (y,x) e R. R is transitive if when (x,y) e R and (y,z) e R then (x,z) e R

11. Discrete Math for CS Understanding Relations as digraphs. If R is a relation represented as a di-graph then R is reflexive if every node has a loop to itself attached. R is symmetric if every directed edge is directed in both directions. R is antisymmetric if there is no bi-directional edge. If there is a directed edge from x to y and another from y to z then there is a directed edge from x to z

12. Discrete Math for CS Understanding Relations as Matrices: Given a binary relation R on a finite set X. Let M be the matrix whose rows and columns are indexed by the elements of X. R is reflexive if the elements on the leading diagonal are all 1(T). R is symmetric if the matrix is symmetric about the main diagonal. R is antisymmetric if there are no symmetrical elements. Hence if mij == 1 the mji != 1. The text says transitivity is not readily apparent. We'll see!

13. Discrete Math for CS Exercise: Evaluate the following relations for the previously mentioned properties: x divides y on the natural numbers x != y on the integers. x is the same age as y in the set of all people. a/b has the same value as c/d in the set of all rationals, Q.

14. Discrete Math for CS The Closure of a Relation: Suppose R is a relation on a set A. Then R A x A. If R does not possess a relation property P, say being reflexive, then there exists another relation R* on A with the three properties R R* R* possesses the desired property. If any other relation, say R', satisfies 1) and 2) then R* R'. We call R* the closure of R with respect to the property P on the set A.

15. Discrete Math for CS Example: A = {1, 2, 3} and R is the relation {(1,1),(1,2),(1,3),(3,1),(2,3)}. Find the closure of R with respect to the reflexive property.

16. Discrete Math for CS Equivalence Relation: A relation R on a set A that is reflexive, symmetric and transitive is said to be an equivalence relation. Equivalence relations generalize the concept of equality. Exercise: Show = is an equivalence relation. Exercise: Show a/b has the same value as c/d in the set of all rationals, Q is an equivalence relation. Exercise: Show �has the same angles� is an equivalence relation on the set of all triangles. Exercise: Show that x R y if xy > 0 for all x, y e R. Exercise: Give a name to the relation R in the previous exercise.

17. Discrete Math for CS Equivalence Relations and Partitions: The examples on the previous slide show there is a strong relationship between equivalence relations and partitions. A partition is a set of non-empty subsets A1, A2, ... of a set A such that A1 U A2 U ... = A Ai Aj = O for every pair i, j where i != j. We call each Ai a block of the partition.

18. Discrete Math for CS Example:

19. Discrete Math for CS Equivalence Class: If R is an equivalence relation on a set E, then

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21. Discrete Math for CS Theorem: Let R be an equivalence relation on a set A, then the equivalence classes of R form a partition of A. Proof: All equivalence classes are non-empty subsets of A. This is because for every x e A, xRx and so x e Ex.

22. Discrete Math for CS Proof (cont): So

23. Discrete Math for CS Proof (cont): We are left having to prove the second part of the definition of a partition. Namely that any two equivalence classes that are not equal are disjoint.

24. Discrete Math for CS Proof (cont): Suppose

25. Discrete Math for CS Example: Let R be a relation on R satisfying xRy iff x � y is an integer. Show R is an equivalence relation and find the equivalence classes � [0], [�] and [sqrt(2)].

26. Discrete Math for CS Partial Orders: A binary relation on a set A that is reflexive, anti-symmetric and transitive is called a partial order and the set is called a partially ordered set with respect to the relation or a poset for short. Partial orders allow elements to �preceed� one another but not necessarily. Examples: subsets of a set are ordered partially

27. Discrete Math for CS Partial Orders: Example: subsets of a set are ordered partially

28. Discrete Math for CS Partial Orders: Example: is a divisor of in the set of natural numbers.

29. Discrete Math for CS Posets: A set on which a partial order is defined is called a poset. If R is a partial order on a set A and xRy we say that x is a predecessor of y and y is a successor of x. An element of A can have many pedecessors but if xRy and there is no z such that xRz and zRy we say x is an immediate predecessor of y (written x ? y). We use a Hasse Diagram to represent immediate predecessors. The vertices of a Hasse Diagram represent the elements of A and if they are directly connected then the lower vertex is the immediate predecessor of the upper vertex.

30. Discrete Math for CS Example: Let A = {1,2,3,6,12,18}. Let R be the relation �is a divisor of�. R is reflexive (a is a divisor of a), anti-symmetric (a is a divisor of b and b is a divisor of a implies a == b) and transitive (a is a divisor of b and b is a divisor of c implies a is a divisor of c). R is a partial order.

31. Discrete Math for CS Total Order: A total order is a partial order in which every two elements of the set are related. Hence if a, b ? A, either aRb or bRa.