Answer to Question #66407 in Abstract Algebra for Preet

Question

Define a relation R on Z , by } R ={( ,n n + 3 k|)k ∈Z.
Check whether R is an equivalence relation or not. If it is, find all the distinct
equivalence classes. If R is not an equivalence relation, define an equivalence
relation on Z . (5)
b) Consider the set }1 X = R {\ − . Define ∗ on X by
X x x x x x x x , x 1
∗ 2 = 1 + 2 + 1 2∀ 1 2 ∈ .
i) Check whether ) ( ,X ∗ is a group or not.
ii) Prove that x ∗ x ∗ x ∗K∗ x (n times) = 1( + )x −1 ∀ n∈N
n
and x ∈X