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Let $R$ be the relation on the natural numbers defined by $(a,b)\in R$ if and only if $2\mid(a+b)$

I already proved this was an equivalence relation, but how do I determine the number of equivalence classes and describe them?

Help is appreciated.

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  • $\begingroup$ Play around with some values. Can you find two integers that are not in the same equivalence class? (Hint: start with simple integers.) What are they? Can you prove that they are in different equivalence classes? $\endgroup$
    – amomin
    Dec 8, 2014 at 22:25

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Just play around with some numbers. Consider $3 \in \mathbb N$. What is it related ("equivalent") to? Well $(3, 5) \in R$, since $2 \mid 8$. But $(3,6) \notin R$, since $2 \not\mid 9$. Continuing, we notice that: $$ 2 \mid (a + b) \iff a + b \text{ is even} \iff a \text{ and } b \text{ have the same parity} $$ where by "parity", I mean whether a natural number is even or odd. So $R$ partitions $\mathbb N$ into two equivalence classes, namely: \begin{align*} [1]_R &= \{1, 3, 5, 7, \ldots\} \\ [2]_R &= \{2, 4, 6, 8, \ldots\} \end{align*}

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Hint:Take the number $1$ to be $a$, what kind numbers are related to $1$ in $R$? Then take $2$, and so on you will get the classes very fast.

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