Prove that a relation `R` on a set `A` is symmetric iff `R=R^(-1)` .

Statement-1: On the set Z of all odd integers relation R defined by (a, b) in R iff a-b is even for all a, b in Z is an equivalence relation. Statement-2: If a relation R on a set A is symmetric and transitive, then it is reflexive and hence an equivalence relation, because (a, b) in Rimplies(b, a)in R" [By symmetry]" (a, b)in R and (b, a) in Rimplies (a,a)in R " [By transitivity]"

Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by (x , y) in R iff x-y is divisible by n , is an equivalence relation on Z.

Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by (x , y) in R iff x-y is divisible by n , is an equivalence relation on Z.

Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by (x , y) in R iff x-y is divisible by n , is an equivalence relation on Z.

Let R = {(a, a)} be a relation on a set A.Then R is

Prove that the relation R in set A = {1, 2, 3, 4, 5} given by R = {(a,b): |a-b| is even} is an equivalence relation .

(i) Prove that the relation ""_(x)R_(y) iff 3 is a factor of (x-y) , defined on the set of integers is an equivalence relation. (ii) If x is the set of real numbers, then prove that the relation R={(a,b): a in x, b in x and a=b} is an equivalence relation.

Show that the relation R on the set A={1,\ 2,\ 3} given by R={(1,\ 2),\ (2,\ 1)} is symmetric but neither reflexive nor transitive.

If R is a symmetric relation on a set A , then write a relation between R and R^(-1) .

If R is an equivalence relation on a set A, then R^-1 is