Prove that the relation R defined on the set Z of integers as `R= {(a, b) : 4` divides `abs(a-b)}` is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 5 divides abs(a-b)} is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 3 divides abs(a-b)} is an equivalence relation.

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

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 .

Show that the relation R defined in the set A of all lines as R = {(L_1, L_2): L_1 and L_2 are parallel lines} is an equivalence relation.

Show that the relation R defined in the set A of all triangles as R={(T_(1),T_(2)):T_(1) is similar to T_(2)} is an equivalence relation.

Prove that the relation R defined in the set A= { x: x in Z, 0 le x le 12} as R = {(a, b) : abs(a-b) . is divisible by 5} is an equivalence relation.

Prove that the relation R defined in the set A= { x: x in Z, 0 le x le 12} as R = {(a, b) : abs(a-b) . is divisible by 3} is an equivalence relation.

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