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Assignment 4, Trimester 3, 2017Discrete mathematicsAssignment 4Institution NameStudent Name

Assignment 4, Trimester 3, 20171.Relations R on a setA={1,2,3,4}Find the matrix representing R and the directed graph corresponding to Ri.R={(1,3),(2,3),(3,1),(3,2),(4,2),(4,4)}R=[0010001011000101]The directed graph will beii.R={(a,b):a2+b2>8(1,1),(1,2),(1,3),(1,4),(2,1),(2,2)(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)¿{(1,3),(1,4),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}R=[0011001111111111]Then the graph isiii.R={(a,b):a−b=0whichmeansa=b}thisgives(1,1),(2,2),(3,3),(4,4)31422143

Assignment 4, Trimester 3, 2017¿hereweobtainthe¿R=[1000010000100001]thenthe graph will be given by2.Relations r on the set of non-negative integersEquivalence relations are reflexive, symmetric and transitivei.R={(a,b):a−bis divisible by 4To be reflexive the set(a,a)∈Rfor all the elements in aa−a=0, which is divisible by 4Testing symmetricIf(a,b)εRthen(b,a)εRFor a set(20,4)thereisanother set (4,20) present inR.Ifa−b=cwhich is divisible by 4 thenb−a=−cwhich also is divisible by 4. R istherefore symmetric.TransitiveA relation R is transitive if whenever(a,b)εR,(b,c)εRthen(a,c)εRAssuming the numbers(27,30)as(a,b)27−3=244=6hencedivisibleby4then(b,c)=(3,27),3−27=−244=−6hencedivisibleby4for this case there must be a number(a,c)=;27−27=04=0whichalsoisdivisibleby4the relation R is thus transitivein conclusion R satisfies all the properties of an equivalence relation.1243

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