Answer to Question #209915 in Discrete Mathematics for Sach

Question

Show that the following relations are Equivalence relations. Find out the Equivalence classes.

(a) ρ = {(a, b) | a – b is an integer} on the set of real numbers R

(b) R = {(a, b) | a = b or a = –b } on the set of integers Z

(c) Congruent modulo 4 relation on Z: R = {(a, b) | a – b is divisible by 4} on set of integers Z

(d) Congruent modulo 5 relation on Z: R = {(a, b) | a – b is divisible by 5} on set of integers Z

(e) R = {(S1, S2) | Length (S1) = Length (S2)} on the set of strings {Si} of English letters

(f) R = {(S1, S2) | If the first 3 bits of S1 and S2 are identical} on the set of all bit-strings {Si} of

length 4

(g) R = {(S1, S2) | If the first 3 bits of S1 and S2 are identical} on the set of all bit-strings {Si} of

length 3 or more