Is this relation symmetric on $\mathbb{Z} \times ( \mathbb{Z} \backslash \{0\}) $

Question

Define a relation $R$ on $\mathbb{Z} \times ( \mathbb{Z} \backslash \{0\})$ by $(a, b) R (x, y)$ iff $ay = bx$.

Checking whether $R$ is symmetric.

$(a,b)R(b,a) \implies a.a = b.b$ which is false for $a = 2$ and $b = 3$.

In the book : Analysis with an introduction to proof, 5th Ed by Steven Lay on Page 65, it is being asked prove that $R$ is an equivalence relation.

I think I have proven $R$ is not an equivalence relation.

Can you verify?

Answer 1

To check that the relation is symmetric: The elements here are of the form $(a,b)\in\mathbb{Z}\times(\mathbb{Z}\setminus \{0\})$. So, you need two elements: $(a,b)$ and $(c,d)$. Therefore, you must show that $$(a,b)R(c,d)\Rightarrow(c,d)R(a,b).$$ Assuming $(a,b)R(c,d)$ means that $ad=bc$. This is, however, (after rearrangement) the same as the relation that you need for $(c,d)R(a,b)$, which is $cb=da$.