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Week 5 ( basic concept of relation )
- 1. BASIC CONCEPT OF RELATIONS Objectives: At the end of the lessons, students are expected/should be able to; 1. Determine the different basic set identities. 2. Identify the different ways of proving set identities. 3. Show the step by step procedure in proving statements using basic set identities. If we want to describe a relationship between elements of two sets A and B, we can use ordered pairs with their first element taken from A and their second element taken from B. Since this is a relation between two sets, it is called a binary relation. Definition: Let A and B be sets. A binary relation from A to B is a subset of AB. In other words, for a binary relation R we have R AB. We use the notation aRb to denote that (a, b)R and aRb to denote that (a, b)R. When (a, b) belongs to R, a is said to be related to b by R. Example: Let P be a set of people, C be a set of cars, and D be the relation describing which person drives which car(s). P = {Carl, Suzanne, Peter, Carla}, C = {Mercedes, BMW, tricycle} D = {(Carl, Mercedes), (Suzanne, Mercedes), (Suzanne, BMW), (Peter, tricycle)} This means that Carl drives a Mercedes, Suzanne drives a Mercedes and a BMW, Peter drives a tricycle, and Carla does not drive any of these vehicles. Functions as Relations You might remember that a function f from a set A to a set B assigns a unique element of B to each element of A. The graph of f is the set of ordered pairs (a, b) such that b = f(a). Since the graph of f is a subset of AB, it is a relation from A to B. Moreover, for each element a of A, there is exactly one ordered pair in the graph that has a as its first element. Conversely, if R is a relation from A to B such that every element in A is the first element of exactly one ordered pair of R, then a function can be defined with R as its graph.
- 2. This is done by assigning to an element aA the unique element bB such that (a, b)R. Relations on a Set Definition: A relation on the set A is a relation from A to A. In other words, a relation on the set A is a subset of AA. Example: Let A = {1, 2, 3, 4}. Which ordered pairs are in the relation R = {(a, b) | a < b}? Solution: R = {(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)} R 1 2 3 4 1 x x x 2 x x 3 x 4 How many different relations can we define on a set A with n elements? A relation on a set A is a subset of AA. How many elements are in AA ? There are n2 elements in AA, so how many subsets (= relations on A) does AA have? The number of subsets that we can form out of a set with m elements is 2m. Therefore, 2n^2 subsets can be formed out of AA. Answer: We can define 2n^2 different relations on A. Let A={a1, a2, ..ak} and B={b1,b2,..bm}. The Cartesian product A x B is defined by a set of pairs {(a1 b1), (a1, b2), … (a1, bm), …, (ak,bm)}. Cartesian product defines a product set, or a set of all ordered arrangements of elements in sets in the Cartesian product Binary Relation Definition: Let A and B be two sets. A binary relation from A to B is a subset of a Cartesian product A x B. Let R A x B means R is a set of ordered pairs of the form (a,b) where a A and b B.
- 3. We use the notation a R b to denote (a,b) R and a R b to denote (a,b) R. If a R b, we say a is related to b by R. Example: Let A={a,b,c} and B={1,2,3}. Is R={(a,1),(b,2),(c,2)} a relation from A to B? Yes. Is Q={(1,a),(2,b)} a relation from A to B? No. Is P={(a,a),(b,c),(b,a)} a relation from A to A? Yes Properties of Relations Definitions: A relation R on a set A is called symmetric if (b, a)R whenever (a, b)R for all a, bA. A relation R on a set A is called antisymmetric if a = b whenever (a, b)R and (b, a)R. A relation R on a set A is called asymmetric if (a, b)R implies that (b, a)R for all a, bA. Example: For the following relations on {1, 2, 3, 4}. 1. R = {(1,1),(1,2),(2,1),(3,3),(4,4)} symmetric 2. R = {(1,1)} symmetric & antisymmetric 3. R = {(1,3),(3,2),(2,1)} antisymmetric & asymmetric 4. R = {(4,4),(3,3),(1,4)} asymmetric Definition: A relation R on a set A is called transitive if whenever (a, b)R and (b, c)R, then (a, c)R for a, b, cA. Example: For the following relations on {1, 2, 3, 4}. 1. R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)} Transitive 2. R = {(1, 3), (3, 2), (2, 1)} Not Transitive 3. R = {(2, 4), (4, 3), (2, 3), (4, 1)} Not Transitive
- 4. Definition (reflexive relation) : A relation R on a set A is called reflexive if (a,a) R for every element a A. Example 1: Assume relation R ={(a b), if a |b} on A = {1,2,3,4} , Is R reflexive? R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} Answer: Yes. (1,1), (2,2), (3,3), and (4,4) A.