Discrete Structure lecture_10

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 # 10TYPES OFRELATIONSKHAWAJA TARIQ MAHMOODLecture # 10TYPES OF RELATIONSREFLEXIVE RELATION:Let R be a relation on a set A. R is reflexive if, and only if, for all a A,(a, a) R. Or equivalently aRa. That is, each element of A is related to itselfREMARKR is not reflexive iff there is an element a in A such that (a, a) R.That is, some element a of A is not related to itself.EXAMPLE:Let A = {1, 2, 3, 4} and define relations R1, R2, R3, R4 on A as follows:R1 = {(1, 1), (3, 3), (2, 2), (4, 4)}R2 = {(1, 1), (1, 4), (2, 2), (3, 3), (4, 3)}R3 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}R4 = {(1, 3), (2, 2), (2, 4), (3, 1), (4, 4)}Then,R1 is reflexive, since (a, a) R1 for all a A.R2 is not reflexive, because (4, 4) R2.R3 is reflexive, since (a, a) R3 for all a A.R4 is not reflexive, because (1, 1) R4, (3, 3) R4DIRECTED GRAPH OF A REFLEXIVE RELATION:The directed graph of every reflexive relation includes an arrow from every point topoint itself (i.e., a loop).theEXAMPLE:Let A = {1, 2, 3, 4} and define relations R1, R2, R3, and R4 on A byR1 = {(1, 1), (3, 3), (2, 2), (4, 4)}Khawaja Tariq Mahmood.Page 2Lecture # 10TYPES OF RELATIONSR2 = {(1, 1), (1, 4), (2, 2), (3, 3), (4, 3)}R3 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}R4 = {(1, 3), (2, 2), (2, 4), (3, 1), (4, 4)}Then their directed graphs are11243R1 is reflexive because atevery point of the

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