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Let R={(m,n):2 divides m+n} on Z. Show t...

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

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Show that if R is an equivalence relation on X then dom R=rngR =X.

Show that if R is an equivalence relation on X, then Dom R = Rng R = X.

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.

If f:X to Y is a function. Define a relation R on X given by R={(a, b): f(a)=f(b)}. Show that R is an equivalence relation on X.

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

Let T be the set of all triangles in a plane with R is a relation in T given by R= {(T_(1),T_(2)): T_(2) is congruent to T_(2) and T_(1), T_(2) in T) . Show that R 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 ~ be defined by (m,n)~(p,q) if mq=np where m, n, p, qinZ -{0}. Show that it 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.

Show that the relation R on the set Z of integers given by R = {(a,b): 2 divides (a - b)} is an equivalence relation.

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

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