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For $$A = \{(−4, −20), (−3, −9), (−2, −4), (−1, −11), (−1, −3), (1, 2), (1, 5), (2, 10), (2, 14), (3, 6), (4, 8), (4, 12)\}$$ define the relation $R$ on $A$ by $(a, b) R (c, d)$ if $ad = bc$.

  • a) Verify that $R$ is an equivalence relation on $A$.
  • b) Find the equivalence classes $[(2, 14)], [(−3, −9)],$ and $[(4, 8)]$.
  • c) How many cells are there in the partition of $A$ induced by $R$?
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What are your thoughts on the problem? What do you know about equivalence relations - do you know what a relation is, and what three properties a relation must satisfy to be an equivalence relations? The more you tell us about what work you have done on this question, the better we can help you. Also, if this is homework, please tag the question accordingly. –  AWertheim Aug 13 '13 at 5:36
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2 Answers

First of all you have to check the property of reflexivity for all elements of $A$. For instance: $-4 \cdot -20= -20 \cdot -4$.

Secondly you have to check all $aRb$ for symmetry. For instance $(-2,-4)R(3,6)$ because $-2 \cdot 6 = -4 \cdot 3$. Now we have to verify if $(3,6)R(-2,-4)$.

Thirdly you have to check the transitivity: if $aRb$ and $bRc$ then $aRc$. For instance: $(3,6)R(4,8)$. We already know that $(-2,-4)R(3,6)$. You now have to check if $(-2,-4)R(4,8)$.

Hints for parts $b$ en $c$ of your question. To get a good intuitive feel for equivalence classes and equivalence relations I suggest you take a piece of paper and draw some dots on it. Each dot will represent an element $a \in A$. Draw a line between two dots $a$ and $b$ if $aRb$. Also draw a line if it turns out $bRa$. Do you know what your drawing should look like when you are finished? Can you see the equivalence classes? How many are there?

Hope this helps

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Leo's answer gives you all that you really need, but it might be helpful to see the computation of one equivalence class; I'll compute $[\langle 4,8\rangle]$. You know that $\langle 4,8\rangle\mathbin{R}\langle c,d\rangle$ if and only if $4d=8c$. This condition is equivalent to the condition $d=2c$, so we're looking for the pairs $\langle c,d\rangle\in A$ such that $d=2c$. The set consisting of precisely those pairs is

$$[\langle 4,8\rangle]=\{\langle -2,-4\rangle,\langle 1,2\rangle,\langle 3,6\rangle,\langle 4,8\rangle\}\;.$$

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