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The relation R in the set Integers given by R = {(a, b) : a – b is divisible by 3} is
  • a)
    reflexive
  • b)
    reflexive but not symmetric
  • c)
    not symmetric and transitive
  • d)
    equivalence relation
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The relation R in the set Integers given by R = {(a, b) : a – b ...
Concept:
1. Reflexive: Each element is related to itself.
R is reflexive if for all x ∈ A, xRx.
2. Symmetric: If any one element is related to any other element, then the second element is related to the first.
R is symmetric if for all x, y ∈ A, if xRy, then yRx.
3. Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third.
R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.
4. R is an equivalence relation if A is nonempty and R is reflexive, symmetric and transitive.
Calculation:
For reflexive:
(a, a) = a – a = 0 is divisible by 3.
For symmetric:
If (a - b) is divisible by 3, then (b – a) = - (a - b) is also divisible by 3.
Thus relation is symmetric.
For transitive:
If (a - b) and (b - c) is divisible by 3
Then (a - c) is also divisible by 3
Thus relation is transitive.
∴ R is an equivalence relation
Hence, option (d) is correct.
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The relation R in the set Integers given by R = {(a, b) : a – b is divisible by 3} isa)reflexiveb)reflexive but not symmetricc)not symmetric and transitived)equivalence relationCorrect answer is option 'D'. Can you explain this answer?
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