Regarding Relations

Asked by thebluebloo | 31st Mar, 2009, 06:51: AM

Expert Answer:

Let a, b c belong to Z

a is related to a , as a-a=0 is divisible by m(reflexive)

if a is related to b  then b must be related to a as a is related to b implies a-b is divisible by m.So b-a must also be divisible by m

since

a-b=mk for some k in Z

then b-a=m(-k)

a-b=rm where r=-k belongs to Z

so ais related to b implies b is related to a((symmetric)

if a is related to b  and b is related to c

 then

 a-b= mk(say, for some k in Z)

 and

b-c= mp(say, for some p in Z)

So a-c=(a-b)+(b-c)=mk+mp=m(k+p) is divisible by m

 thus a-c is divisible by m

i.e. a is congruent to c modulo m

i.e. a is related to c  (transitive)

hence the answer.

Answered by  | 31st Mar, 2009, 08:53: AM

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