Consider the equivalence relation $\sim$ on $\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$ defined by $(a,b) \sim (c,d)$ if $a \cdot d = b \cdot c$. Describe the equivalence classes in terms of familiar mathematical objects?

The above is a homework problem. I can see some patterns in the equivalence classes, however I'm not sure how to answer or approach the question given. Thanks!

  • $\begingroup$ Have you written down some examples of equivalent pairs? What do they remind you of? Can you describe the set of all pairs equivalent to (1,2) for example? $\endgroup$ – G Tony Jacobs Jul 7 '14 at 12:35
  • $\begingroup$ [((1,1),(2,2)),((1,2),(1,2)),((2,1),(2,1)),((2,2),(1,1))]. They look like a square grid of points. Is the answer they are looking for a square? $\endgroup$ – clay Jul 7 '14 at 12:37
  • $\begingroup$ No, you just want pairs that are equivalent to $(1,2)$. For example, you have $(1,2)\sim (1,2)$, $(1,2)\sim (2,4)$, $(1,2)\sim (3,6)$, etc. Do you see why those are true? $\endgroup$ – G Tony Jacobs Jul 7 '14 at 12:39
  • $\begingroup$ $(1,1)\not\sim (2,1)$, as you seem to be suggesting, since $2\neq 1$. $\endgroup$ – HSN Jul 7 '14 at 12:39
  • $\begingroup$ @HSN, I think the OP was listing several examples of equivalent pairs, but not saying that they're all equivalent to each other. Look at the use of parentheses. He's writing down part of the equivalence relation as a set of ordered pairs. $\endgroup$ – G Tony Jacobs Jul 7 '14 at 12:42

The equivalence classes corespond to rational numbers with $\frac{a}{b}$ representing the class $(a,b)$.


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