Answer to Question #311158 in Discrete Mathematics for Gurleen

Question #311158

Give an example of relation which is Reflexive, anti symmetric & transitive but not symmetric

1
Expert's answer
2022-03-14T18:36:00-0400

Let N\N be the set of all natural numbers. Let R be a relation defined on N\N as aRb    a divides ba \text{R} b \iff a \text{ divides } b.


Since a divides itself for all aNa \in \N, R is reflexive.


For if aRb and bRca\text{R}b \text{ and } b\text{R}c, then aRca\text{R}c.

To prove the above statement, we haveaRb    a divides b    b=ma for some positive integer mbRc    b divides c    c=nb for some positive integer nUsing b=ma, we get c=mna    a divides c    aRca\text{R}b \iff a \text{ divides } b \implies b = ma \text{ for some positive integer m}\\ b\text{R}c \iff b \text{ divides } c \implies c = nb \text{ for some positive integer n}\\ \text{Using } b = ma, \text{ we get } c = mna \implies a \text{ divides }c \implies a\text{R}c

Hence R is transitive.


Now, if aRb and bRaa\text{R}b \text{ and } b\text{R}a then b=ma and a=nb    a=mna    mn=1    m=n,b= ma \text{ and }a = nb \implies a = mna \iff mn = 1\iff m=n, since m and n are integers. This proves R is antisymmetric.


R is not symmetric, because 4R84\text{R}8, but 8̸R4.8\not\text{R}4.


Thus we have shown a relation R which is reflexive, transitive, antisymmetric but not symmetric.


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