Let ~ be defined by (m,n)~(p,q) if mq=np where m, n, p,`qinZ`-{0}. Show that it is an equivalence relation.

Let R={(m,n):2 divides m+n} on Z. Show that R is an equivalence relation on Z.

Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R (u, v), if and only if xv = yu. Show that R is an equivalence relation.

Let R be the relation on Z defined by aRb iff a-b is an even integer. Show that R is an equivalence relation.

If N denotes the set of all natural numbers and R be the relation on N xx N defined by (a, b) R (c, d) if ad(b+ c)=bc(a+d) . Show that R is an equivalence relation.

The ralation R on Z is defined by for m, nin Z , mRnimpliesm/n is a power of 2. Examine whether it is an equivalence relation.

Show that the relation ~ on ZZ-{0}timesZZ-{0} defined by (m,n)~(p,q)Leftrightarrowmq=np is an equivalence relation.

Show that the relation R on Z – {0} defined by R={(m,n)|frac[m][n] is power of 5} is an equivalence relation.

Let n be a fixed positive integer. Define a relation R on Z as follows for all a, b in Z, aRb, if and only if a-bis divisible by n. Show that R is an equivalence relation.

Let R be a relation defined on the set of natural numbers N as follows: R = {(x,y): x in N,y in N and 2x + y = 24). Then, find the domain and range of the relation R. Also, find whether R is an equivalence relation or not.

Show that the relation R on the set A of points in a plane given by R ={(P,Q): distance of the point P from the origin is same as the distance of the point Q from the origin) is an equivalence relation. Further, show that the set of all points related to a point P ne (0,0) is the circle passing through P with origin as centre.