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Let $P$ be the set of all primes. Define a relation in $P$ by $x\sim y$ if $x+y$ is even. It turns out that $\sim$ is an equivalence relation. Calculate the total number of equivalence classes.

The only even prime is 2, so we can consider all the cases without 2 such that the sum of 2 prime numbers must be even.

Parcly Taxel
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Rot Civ
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  • i realize that the only even prime number is 2. So we can just simply consider all the cases without 2 such that the sum of 2 prime numbers must be even. – Rot Civ Oct 04 '16 at 03:42
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    Next time just add your attempts into the post, I did it for you already for this question. – suomynonA Oct 04 '16 at 03:43

1 Answers1

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The sum of two numbers is even if either, the two numbers are both even, or they are both odd.

$2$ is the only even prime.   So the only prime to which $2$ can be added to yield an even number, is itself.   Thus $\{2\}$ is one equivalence class. $${[2]}_\sim= \{2\}$$

$3$ is an odd prime number.   What primes are there to which adding $3$ yields an even number?   That gives you the equivalence class of ${[3]}_\sim$.

Now, take a prime that is not in either ${[2]}_\sim$ or ${[3]}_\sim$, and repeat the process until you run out of primes.

Already done?   That was quick, wasn't it.

$${{2}, {3,5,7,11,\ldots}}$$

Graham Kemp
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