Assignment 4, Trimester 3, 2017Discrete mathematicsAssignment 4Institution NameStudent Name
Assignment 4, Trimester 3, 20171.Relations R on a set A={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]then the graph will be given by2.Relations r on the set of non-negative integers Equivalence relations are reflexive, symmetric and transitivei.R={(a,b):a−bis divisible by 4To be reflexive the set (a,a)∈R for all the elements in a a−a=0, which is divisible by 4Testing symmetricIf (a,b)εR then (b,a)εRFor a set (20,4)thereisanother set (4,20) present in R.If a−b=cwhich is divisible by 4 then b−a=−c which also is divisible by 4. R is therefore 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=6hencedivisibleby4 then (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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