Math CentralQuandaries & Queries


Question from syed, a student:
D is the relation defined on Z as follows:
Z is an integer
For all m,n E Z, m D n if and only if 3 l (m^2 - n^2).
Find out whether the above relation satisfies to reflexive , Symmetric & transitive?
Is it an equivalence?


First off, it looks as if your notation is a bit mangled in the email. I'm assuming you mean

for all integers m,n, m D n if and only if 3 divides (m2 -n2)

Is it reflexive?
<=>is it always true for (m,m)?
<=> does 3 divide (m2-m2)?

Is it symmetric?
<=>if it is true for (m,n) is it true for (n,m)?
<=>if 3 divides (m2-n2) does .....

Is it transitive?
<=>if it is true for (l,m) and for (n,m) is it true for (l,n)?
<=>if 3 divides (l2-m2) and ......

If all of these are true it's an equivalence. Otherwise it's not.

You should be able to answer each of these questions on your own.

And now you've seen how it's done you should be able to set up problems like this for solution on your own, too.

Good Hunting!


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