GO Classes Set Theory And Algebra Practice Set 1 | Question: 7

Question

Define $\mathcal{R}$ the binary relation on $\mathbb{N} \times \mathbb{N}$ to mean $(a, b) \mathcal{R}(c, d)$ iff $b \mid d$ and $a \mid c$

Answer 1

Let's analyze the properties of the given relation R:

 

Symmetry: For a relation to be symmetric, if (a, b) is related to (c, d), then (c, d) must also be related to (a, b). In this case, if b/d and a/c, it does not imply that d/b and c/a. Therefore, the relation is not symmetric.

 

Reflexivity: For a relation to be reflexive, every element must be related to itself. In this case, for the relation (a, b)R(a, b) to hold, we need b/b and a/a, which implies b = b and a = a, which is true. Therefore, the relation is reflexive.

 

Transitivity: For a relation to be transitive, if (a, b) is related to (c, d), and (c, d) is related to (e, f), then (a, b) must be related to (e, f). In this case, if b/d and c/e, it does not imply that b/e. Therefore, the relation is not transitive.

 

 as the relation R is reflexive but not symmetric or transitive.