Prove that the following relation R in Z of integers is an equivalence relation : `R = [ (x, y) : x - y` is an integer}

Prove that the following relation R in Z of integers is an equivalence relation : R = [ (x, y) : 2x - 2y is an integer}

Prove that the following relation R in Z of integers is an equivalence relation : R = [ (x, y) : 3x- 3y is an integer}

If relation R defined on set A is an equivalence relation, then R is

Prove that the relation R in Z of integers given by: R = {(x, y): 3x - 3y is an integer } is an equivalence relation. [3] If f: RrarrR defined by f(x) = (6- 5x)/7 is an invertible function, find f^-1 .

Prove that the relation R in Z of integers given by: R = {(x, y): 2x - 2y is an integer } is an equivalence relation. [3] If f: RrarrR defined by f(x) = (4- 3x)/5 is an invertible function, find f^-1 .

Prove that the relation R in Z of integers given by: R = {(x, y): x - y is an integer } is an equivalence relation. [3] If f: RrarrR defined by f(x) = (3- 2x)/4 is an invertible function, find f^-1 .

Determine whether each of the following relations are reflexive, symetric and transitive Relation R in the set Z of all integers defined as R = (x,y) : x-y is an integer.

Determine whether each of the following relations are reflexive, symmetric and transitive : (i) Relation R in the set A = {1, 2, 3,......, 13, 14} defined as R={(x,y): 3x-y=0} (ii) Relation R in the set N of natural numbers defined as R={(x,y): y=x+5 " and " x lt 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y): y is divisible by x}. iv) Relation R in the set Z, of all integers defined as R = {(x, y) : x -y is an integer}.

Prove that on the set of integers. Z, the relation R defined as aRb hArr a = +-b is an equivalence relation .