I was tasked with this question:
Let $X = \{0,1,2,3,4\}$. Define a relation $R$ on $X$ such that $xRy$, if $x + y = 4$.
I don't understand/know what syntax I should use to define this as a relation. What are they asking when they say "$x R y$, if $x + y = 4$"? What are $x$ and $y$ here? Does it mean $x$ is from set $X$? If that's the case, where is $y$ from? There certainly is no set $Y$ for $y$.
To narrow down what I'm asking: how do I define this relation? I'd appreciate it if you could walk me through the steps. Also, I'm only a month into this subject of Discrete Math, and I'm having difficulty, and I've only learned up to this point about proofs, relations, functions, sets, methods of proof, relations and equivalences, and predicate logic. I know these things well enough, just try not to use any logic that is beyond this scope.
We have a set $X = \{0,1,2,3,4\}$. When we say "a relation $R$ on $X$", we mean that the relation consists of pairs $(x,y) \in X \times X$, in other words, the elements that make up the pairs of the relation $R$ comes from the set $X$.
So for this specific relation $R$, what is the requirement for being an element of it? Well, the question says that $x R y$ (which we can also write $(x,y) \in R$) if $x + y = 4$. That is, the elements of our relation are pairs $(x,y) \in X \times X$ that satisfy the condition that $x + y = 4$. So the relation $R$ on $X$ is given by $$R = \{(x,y) \in X \times X \; | \; x + y = 4\}.$$ Looking at the elements of $X$, we see that the following pairs satisfy this condition. $$R = \{(0,4),(4,0),(1,3),(3,1),(2,2)\}.$$