Solution

The 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.