Question

Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

A. Symmetric only
B. Reflexive only
C. Transitive only
D. An equivalence relation

Answer 1

To show that the relation R is an equivalence relation, we need to establish that it is reflexive, symmetric, and transitive. Reflexivity can be proved by substituting a for both a and b and verifying the equation.

Symmetry can be proved by interchanging the roles of a, b, c, and d and verifying the equation. Transitivity can be proved by substituting the given relations for (a, b) and (b, c) in the equation and verifying it simplifies to the equation for (a, c) in R.

Step-by-step explanation:

To show that the relation R is an equivalence relation, we need to establish that it is reflexive, symmetric, and transitive.

To show reflexivity, we need to prove that for every natural number a, (a, a) is in R. This can be done by substituting a for both a and b in the given relation and verifying that the equation holds true.

To show symmetry, we need to prove that for any two natural numbers a and b, if (a, b) is in R, then (b, a) is also in R. This can be done by interchanging the roles of a, b, c, and d in the given relation and verifying that the equation still holds true.

To show transitivity, we need to prove that for any three natural numbers a, b, and c, if (a, b) is in R and (b, c) is in R, then (a, c) is also in R. This can be done by substituting the given relations for (a, b) and (b, c) in the equation and verifying that it simplifies to the equation for (a, c) in R.