Discrete Structure lecture_16

Apr 28th, 2015
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It includes all the details about the Discrete structure with examples.Step by step solution.

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DISCREAT STRUCTURELECTURE # 12FUNCTIONKHAWAJA TARIQ MAHMOODLecture # 12FUNCTIONSRELATION AND FUNCTIONS:A function F from a set X to a set Y is a relation from X to Y that satisfiesthe following two properties1. For every element x in X, there is an element y in Y such that (x,y) F.In other words every element of X is the first element of some ordered pair of F.2. For all elements x in X and y and z in Y, if (x,y) F and (x,z) F, then y = zIn other words no two distinct ordered pairs in F have the same first element.EXAMPLE:Which of the relations define functions from X = {2,4,5} to Y={1,2,4,6}.a.R1 = {(2,4), (4,1)}b.R2 = {(2,4), (4,1), (4,2), (5,6)}c.R3 = {(2,4), (4,1), (5,6)}SOLUTION :a. R1 is not a function, because 5 X does not appear as the first element in any orderedpair in R1.b. R2 is not a function, because the ordered pairs (4,1) and (4,2) have the same first elementbut different second elements.c. R3 defines a function because it satisfy both the conditions of the function that is everyelement of X is the first element of some order pair and there is no pair which has thesame first order pair but different second order pair.EXERCISE:Let A = {4, 5, 6} and B = {5, 6} and define binary relations R and S from A to Bas follows:for all (x,y) A B, (x,y) R xyfor all (x,y) A B, xSy 2|(x-y)a. Represent R and S as a set of ordered pairs.Khawaja Tariq Mahmood.Page 2Lecture # 12

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