1 X is the set of real-valued functions of a real variable whose domain contains the point 2 and R is the relation defined by fRg if and only if f(2) = g(2). Is this relation transitive? Explain.
For this one am I suppose to look at one side and fix a function (for example fix f(x)=x^2) to that side and then show if it is transitive or not?
2 Fix an $\epsilon > 0.$ $X$ is the set of real valued functions of a real variable whose domain is all of $R$ and $R$ is the relation defined by $fRg$ if and only if $\left |f(x)-g(x)\right |$ $< \epsilon$ for each $x \in R$. Is this relation transitive? Explain.
I think this one is not transitive because the fix $\epsilon > 0$ will cause some problems, but I don't know how I should explain that.
For the first question, let $f,g,h \in X$, if $f(2)=g(2)$ and $g(2)=h(2)$, can you conclude that $f(2)=h(2)$?
For the second question, let $f(x) = -\frac{\epsilon}2$, $g(x)=0$, $h(x)=\frac{\epsilon}2$, try to verify that this is a counterexample.