GATE CS 2019 Set 1 — Question 20
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Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields
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Question
Let be an arbitrary group. Consider the following relations on :
if and only if such that
if and only if
Which of the above is/are equivalence relation/relations?
if and only if such that
if and only if
Which of the above is/are equivalence relation/relations?
Correct answer
(B) R₁ only
Solution
is the conjugacy relation, which is a known equivalence relation. Proof:
1.Reflexive: , so . (True)
2.Symmetric: If , then , so . (True)
3.Transitive: If and , then , so . (True)
is defined by .1.Reflexive: . This is not true for all groups (only for groups where every element is its own inverse). Thus, is not reflexive in general.
Therefore, only is an equivalence relation.Continue learning with Success Tracker
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