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Relation R is defined as
R = {(a, b) | (a - b) = km for some fixed integer m and a, b, k ∈ z}, then R is
  • a)
    Reflective but not symmetric
  • b)
    Symmetric but not transitive
  • c)
    Transitive but not reflective
  • d)
    An equivalence relation
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Relation R is defined asR = {(a, b) | (a - b) = km for some fixed inte...
Given:
Relation R is defined by function R = {(a, b) | (a - b) = km for some fixed integer m and a, b, k ∈ z}.
  1. For the same number a, R = a - a = 0 × m, where m is a fixed integer and 0 є z, hence the relation R is reflexive.
  2. For two numbers (a, b) if R = a - b = km, where m is a fixed integer and k є z, then for (b, a), R = b - a = -(a - b) = -km, where -k є z. Hence relation R is symmetric.
  3. Consider three numbers a, b, and c. If, for (a, b), R = a - b = km and for (b, c), R = b - c = lm, where m is a fixed integer and k, l є z, then for (a, c), R = a - c = a - b + b - c = km + lm = m(k + l), where (k + l) є z, since k, l є z. Hence relation R is transitive.
Hence, relation R is an equivalence relation.
Hence, the correct answer is option 4.
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Relation R is defined asR = {(a, b) | (a - b) = km for some fixed integer m and a, b, k ∈ z}, then R isa)Reflective but not symmetricb)Symmetric but not transitivec)Transitive but not reflectived)An equivalence relationCorrect answer is option 'D'. Can you explain this answer?
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