Let m be fixed positive integer. Tow integers a amd are said to be congruent modulo m, written a=b (modm) if m divides a- b .Show that the relation of congruent modulo m is an equivalence relation .

Let Z Z be the set of all integers and let m be an arbitrary but fixed positive integer. Show that the relation "congruence modulo m" on Z Z defined by : a -= b '(mod m)' implies (a-b) is divisible by m, for all a,b in Z Z is an equivalence relation on Z Z .

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

Let R be the set of real numbers Statement-1: A = {(x,y) in R xx R : y-x is an integer} is an equivalence relation on R. Statement -2 : B = {(x,y) in R xx R : x= alpha y for some relational number alpha } is an equivalence relation on R.

Let Z be the set of all integers and R be the relation on Z defined as R={(a , b); a ,\ b\ in Z , and (a-b) is divisible by 5.} . Prove that R is an equivalence relation.

Let R be the relation over the set of integers such that mRn if and only if m is a multiple of n. Then R is

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 m and n are positive integers and m ne n , show : int_(0)^(pi)cosmxcosnxdx=0

Let T be the set of all triangles in a plane with R a relation in T given by R ={(T _(1) , T _(2)): T _(1) is congruent to T _(2) } Show that R is an equivalence relation.

m is a positive integer. If ((1+i)/(1-i))^m=1 ,then least value of m will be

If m is an even integer, show that m^(2) is also an even integer.