COMEDK2024Evening ShiftMathematicsSets and RelationsActual
Which of the following relations on the set of real numbers R is an equivalence relation?
Options
- Aa R₄ b a < b
- Ba R₂ b a b
- Ca R₁ b |a|=|b|
- Da R₃ b adivides b
Correct answer
C. a R₁ b |a|=|b|
Step-by-step solution
A relation R is an equivalence relation if it is reflexive, symmetric, and transitive. For option (1), a R₄ b a For option (2), a R₂ b a b . This is not symmetric because 2 1 is true, but 1 2 is false. For option (3), a R₁ b |a| = |b| . Reflexivity: |a| = |a| is true for all a R , so a R₁ a . Symmetry: If |a| = |b| , then |b| = |a| , so a R₁ b b R₁ a . Transitivity: If |a| = |b| and |b| = |c| , then |a| = |c| , so a R₁ b and b R₁ c a R₁ c . Since all three conditions are satisfied, this is an equivalence relation.