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n the set N×N, the relation R is defined by (a, b) R(c,d)⇔ad=bc. Then R ispartial order relationequivalence relationreflexive and transitive but not symmetricsymmetric and transitive but not reflexive

Question

n the set N×N, the relation R is defined by (a, b) R(c,d)⇔ad=bc. Then R ispartial order relationequivalence relationreflexive and transitive but not symmetricsymmetric and transitive but not reflexive

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Solution

The relation R is defined as (a, b) R(c,d) if and only if ad = bc. We can check the properties of this relation to determine what type of relation it is.

  1. Reflexive: A relation is reflexive if (a, a) R (a, a) for all a. In this case, aa = aa, so the relation is reflexive.

  2. Symmetric: A relation is symmetric if for all (a, b), if (a, b) R (c, d) then (c, d) R (a, b). In this case, if ad = bc, then it is also true that cb = da, so the relation is symmetric.

  3. Transitive: A relation is transitive if for all (a, b) and (c, d), if (a, b) R (c, d) and (c, d) R (e, f) then (a, b) R (e, f). In this case, if ad = bc and ce = df, then it is also true that ae = bf, so the relation is transitive.

Therefore, the relation R is an equivalence relation, because it is reflexive, symmetric, and transitive.

This problem has been solved

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