I have a problem with this task. If anyone had a similar problem it would help me.
The task is:
In the set $S={(-2,-1,0,1,2,3)}$ the relation is defined $\rho$ with $(\forall t,m \in S) t \rho m \iff 2tm\ge t+m$. Examine the properties of the relation $\rho$. If an equivalence relation is given find all classes.
I tried this:
Reflexivity:
$(\forall t\in S) t\rho t$
$2t^{2}\ge2t$ Reflexivity is satisfied
Symmetric:
$(\forall t,m\in S) t\rho m \iff m\rho t\\2tm\ge t+m \iff t+m \le2tm$
The symmetry is satisfied
And now I don’t know how to prove transitivity ?
Thanks in advance !
To show transitivity, you need to show that if $t,m,n\in S$, $t\rho m$, and $m\rho n$, then $t\rho n$. In other words, you must show that if $2tm\ge t+m$ and $2mn\ge m+n$, then $2tn\ge t+n$. That looks rather messy, so you might first try finding a counterexample to transitivity, but if you do try, you’ll not find one. At that point you could go back and tinker with the algebra to try to prove transitivity, but the small size of $S$ makes a much simpler approach feasible: you can simply go through and check all $6^2=36$ pairs of elements of $S$ to see which ones are related by $\rho$. Better yet, since you’ve already shown that $\rho$ is reflexive and symmetric, you need check only the $15$ pairs $\langle t,m\rangle$ with $t<m$. If you do this, you should get the following table, where the numbers in the first column are possible values of $t$, the numbers across the top are possible values of $m$, and a $+$ in row $t$, column $m$ indicates that $t\rho m$.
$$\begin{array}{r|cc} &-2&-1&0&1&2&3\\\hline -2&+&+&+&-&-&-\\ -1&+&+&+&-&-&-\\ 0&+&+&+&-&-&-\\ 1&-&-&-&+&+&+\\ 2&-&-&-&+&+&+\\ 3&-&-&-&+&+&+ \end{array}$$
This table shows that $t\rho m$ if and only if either $t\le 0$ and $m\le 0$, or $t>0$ and $m>0$. Now use that information either to show that $\rho$ is transitive or to find a counterexample to transitivity.