Let R be a relation from Q into Q defined by R ={(a, b) :a, b€Q and a-b€Z}.
Show that
1.(a,a)€R for all a€Q
2(a,b)€R implies (b, a) €R
3(a,b) €R, (b, c) €R implies (a, c) €R

Dear student,

Please find below the solution to the asked query:

1. To show: For all aQ, a,aRLet aQ, that a is any rational number.Then we have   a-a=0Z  0 is an integerThen by definition of relation R, it implies that  a,aR2. To show:  a,bRb,aRLet a,bR, so that  a,bQThis gives,   a-bZNote that if xZ, then its negative -xZThis implies that,     - a-bZ    -a+bZ      b-aZThen by definition of relation R, it implies that  b,aR3. To show:  a,bR, b,cRa,cRLet a,bR, b,cR so that  a,b,cQThis gives,   a-bZ , b-cZSince Z is closed under addition.This implies that,      a-b+b-cZ    a-b+b-cZ      a-cZThen by definition of relation R, it implies that  a,cR

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