Let R be a relation on \mathbb{Z} defined by R=\{(p,q)\in\mathbb{Z}\times\mat



Answered question


Let R be a relation on Z defined by
R={(p,q)Z×Zpq is a multiple of 3}
a) Show that R is reflexive.
b) Show that R is symmetric.
c) Show that R is transitive.

Answer & Explanation



Skilled2021-08-20Added 118 answers

a) For any element pZ
is a multiple of 0.
Thus, (p,p)RR is reflexive.
b) Let (p,q)R Then
pq is a multiple of 3, that is pq=3k for some kZ
To show that (q,p)R Consider
where m=kZ
So qp is a multiple of 3
R is symmetric
c) Let (p,q)R and (q,r)R Then both pq and qr are multiples of 3, that is,
pq=3k,kZ and qr=3m,mR
To show (p,r)R. For that, consider
pr is a multiple of 3.
Thus, R is transitive.

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