Prove an equivalence relation: $mRn \Leftrightarrow 3m-5n$ is even

discrete-mathematics relations equivalence-relations

Solution 1:

We have $$(3m - 5n) + (3n - 5m) = -2m - 2n =-2(m+n) $$ which is even. So the first term of our sum is even if and only if the second term is.

Solution 2:

My suggestion would be to rephrase the definition first. A sum (difference) of two integers is even iff the two integers have the same parity, i.e. they are both odd or both even.

So $3 m - 5 n$ is even if

  • $3 m$ and $5 n$ are both odd, that is, $m$ and $n$ are both odd, or
  • $3 m$ and $5 n$ are both even, that is, $m$ and $n$ are both even.

With this reformulation, it should be straightforward to prove that $R$ is an equivalence relation, and to determine the equivalence classes.

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