Which of the following equivalent relation of a group G?
R 1 : ∀ a , b ∈ G , a R 1 b if only ∃ g ∈ G : a = g − 1 bg
R 2 : ∀ a , b ∈ G , a R 2 b if only a = b –1
- Both R 1 and R 2
- R 1
- R 2
- None of these
Which of the following equivalent relation of a group G?
R 1 : ∀ a , b ∈ G , a R 1 b if only ∃ g ∈ G : a = g − 1 bg
R 2 : ∀ a , b ∈ G , a R 2 b if only a = b –1
Given R1 is a equivalence relation, because it satisfied reflexive, symmetric, and transitive conditions:
aRb ⇒ a = g–1bg for some g ⇒ b = gag–1 = (g–1)–1ag–1 g–1 always exists for every g ∈ G.
aRb and bRc ⇒ a = g1–1bg1 and b = g2–1 cg2 for some g1g2 ∈ G. Now a = g1–1 g2–1 cg2g1 = (g2g1)–1 cg2g1 g1 ∈ G and g2 ∈ G ⇒ g2g1 ∈ G since group is closed so aRb and aRb ⇒ aRc
R2 is not equivalence because it does not satisfied reflexive condition of equivalence relation:
aR2a ⇒ a = a–1 ∀a which not be true in a