The relation R defined in the set N of natural number as `AAn,minN` if on division by 5 each of the integers n and m leaves the remainder less than 5. Show that R is equivalence relation. Also obtain the pairwise disjoint subset determined by R.

The relation R defined on the set of natural numbers as {(a,b) : a differs from b by 3} is given by

The relation R is defined on the set of natural numbers as {(a,b): a = 2b}, the R^(-1) is given by

R is relation in N xxN as (a,b) R (c,d) hArr ad = bc . Show that R is an equivalence relation.

Prove that the relation R defined on the set N of natural numbers by xRy iff 2x^(2) - 3xy + y^(2) = 0 is not symmetric but it is reflexive.

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 n be a fixed positive integer. Define a relation R on Z as follows: (a , b)R a-b is divisible by ndot Show that R is an equivalence relation on Zdot

If R_1 and R_2 are equivalence relations in a set A, show that R_(1) cap R_2 is also an equivalence relation.

Let N denote the set of all natural numbers and R be the relation on NxN defined by (a , b)R(c , d) a d(b+c)=b c(a+d)dot Check whether R is an equivalence relation on NxNdot

Let R be the realtion defined in the set A = {1,2,3,4,5,6,7} by R ={(a,b): both a and b are either odd or even}. Show that R is an equivalance relation. further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all elements of subset {2, 4, 6} are related to each other, but no element of the subset {1,3,5,7} is related to any element of the subset {2,4,6}.

Let R be relation defined on the set of natural number N as follows : R = {(x,y) : x in N, y in N , 2x + y =41} . Find the domian and range of the relation R . Also verify whether R is reflexive, symmetric and transitive.