Can a relation be defined outside its given set? Also I don't quite understand the second question.

Question

1. Consider the relation $R$ on the set $\{1,2,3\}$ where:
   $xRy$ if any only if $x = 5$.
   What is the relation $R$ as a set?
   

2. Let $R$ be the relation on the set of integers, s.t., $aRb$ if and only if $a=b$ or $a= -b$.
   Is $R$ an equivalence relation?

Answer 1

Writing the first one more formally, we get $$R =\{(x,y)\mid x,y\in\{1,2,3\} \land x=5\} = \emptyset$$. But I must admit that the comparison of an element $x$ of the set $\{1,2,3\}$ to the element 5 outside of that set may appear to be a bit weird.

For the second one consider the defining axioms of an equivalence relation. Clearly the relation is reflexive $(x,x)\in R$ and symmetric $a=-b \Leftrightarrow b=-a$. But is it transitive? I.e. does $aRb \land bRc \implies aRc$ hold?

Answer 2

The first question just checks if you understand the definitions. What is a relation $R$ on a set $X$? By definition this is just a subset of the cartesian product $X\times X$. In your example $X=\{1,2,3\}$. So take any $(x,y)\in X\times X$ and let's check if it belongs to $R$. It belongs to $R$ if and only if $x=5$. But that can't be the case, since $x\in\{1,2,3\}$. So the pair $(x,y)$ does not belong to $R$. Hence $R=\emptyset$.

What exactly you don't understand about the second question? Do you know what is an equivalence relation?