Question
Download Solution PDFLet N denote the set of all natural numbers and R be the relation on NXN defined by (a, b) R(c, d), if ad(b + c) = bc(a + d). Then R is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe correct answer is An equivalence relation
Key Points We are given a relation R defined on pairs of natural numbers, where (a, b) R (c, d) if ad(b + c) = bc(a + d).
We can check whether R is symmetric, reflexive, transitive, or an equivalence relation by verifying the condition for each .
- Symmetry:
- For a relation to be symmetric, if (a, b) R (c, d) then (c, d) R (a, b).
- Say we have (a, b) R (c, d), then we have ad(b + c) = bc(a + d).
- Swap a and b with c and d, then we get: cd(a + b) = ab(c + d) which is the original formula, therefore R is symmetric.
- Reflexivity:
- For a relation to be reflexive, (a, b) R (a, b) must be true for all a, b.
- Let's plug (a, b) into the formula: ab(a + b) = ba(a + b) which is always true. Therefore R is reflexive.
- Transitivity:
- For a relation to be transitive, if (a, b) R (c, d) and (c, d) R (e, f), then (a, b) R (e, f).
- However, checking the transitive property for this expression is
- ab/(a - d) = cd/(c - d) and cd/(c - d) = ef/(e - f)
- ab/(a - d) = ef/(e - f)
- af(b + e) = be(a + f)
- (a, b) R (e, f)
- Hence ,R is transitive
Above question follow all condition so it is an equivalence relation.
Last updated on Jun 6, 2025
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