Q. Let 𝐺 be an arbitrary group. Consider the following relations on 𝐺:
𝑅1: ∀𝑎, 𝑏 ∈ 𝐺, 𝑎 𝑅1𝑏 if and only if ∃𝑔 ∈ 𝐺 such that 𝑎 = 𝑔−1𝑏𝑔
𝑅2: ∀𝑎, 𝑏 ∈ 𝐺, 𝑎 𝑅2𝑏 if and only if 𝑎 = 𝑏−1
Which of the above is/are equivalence relation/relations?
(A) 𝑅1 and 𝑅2
(B) 𝑅1 only
(C) 𝑅2 only
(D) Neither 𝑅1 nor 𝑅2
Ans: 𝑅1 only
Solution:
