Question

(10) Define a relation R on Zn (the integers mod n) as follows: lal isR related to [b (i.e. [an Rbn) iff there is [cn E Gn such that a b (a) Show that R is an equivalence relation on Zn (b) Give all the equivalence classes for R when n-12.

Answer 1

Reflexive [a]_n R [a]_n since [1]_n elementof G_n such that [a]_n = [a]_n [1]_n therefore R is reflexive Let [a]_n R [b]_n i.e [a]_n elementof G_n such that [a]_n = [b]_n [c]_n [b]_n = [a]_n [c]^-1_n [b]_n = [a]_n [K]_n where [K]_n = [c]^-1_n therefore [b]_n R [a]_n therefore R is symmetric Let [a]_n R [b]_n & [b]_n R [d]_n [a]_n = [b]_n [a]_n & [b]_n = [d]_n [C_2]_n So [a]_n = [d]_n [C_1]_n [C_2]_n [a]_n = [d]_n [K]_n [a]_n R [d]_n therefore R is Transitive therefore R is an equivalence relation Now but to find the equivalence class [[a]_n] = {[b]_n [a]_n R [b]_n rightarrow [a]_n = [b]_n [c]_n} [[1]_12] = {-1, 1} [[2]_12] = {-7, -2, -1, 2, 7} [[3]_12] = { -3, -1, 1, 3, 5, 9, 11,} [[4]_12] = {-10, -7, -4, -1, 1, 4, 7, 10} [[5]_12] = {-5, -1, 1, 5,} [[6]_12] = {-9, -7, -5, -3, -6,