im working on the practice problem on unit about sets and relations The question is: Let a = {1,2,3,4} and R be a binary relation on A x A given by: ((a,b),(c,d)) ∈R if and only if a divides c and b divides d. Show that R is an order and draw its diagram.
I know that in order for relation to be an order, it has to be reflexive,antisymmetric and transitive. However, im having difficulty proving if its reflexive,antisymmetric and transitive in this case..
Thanks in advance!
We can prove that it is reflexive, for example, as follows:
$R$ is reflexive if for every element $(a,b) \in A \times A$, we have $((a,b),(a,b)) \in R$. That is, $R$ is reflexive if for every pair $(a,b)$ of numbers from $1$ to $4$, $a$ divides $a$ and $b$ divides $b$.
Is it always true that $a$ divides $a$? That $b$ divides $b$? If so, then $R$ is reflexive.