If $X = \{1,2,3,4,5\}$ and $Y = \{1,3,5,7,9\}$, determine which of the following sets represent a relation? Question: If $X = \{1,2,3,4,5\}$ and $Y = \{1,3,5,7,9\}$, determine which of the following sets represent a relation?
Options:
A) $R_1=\{(x,y):y=x+2,x\in X,y\in Y\}$;
B) $R_2=\{(1,1),(1,2),(3,3),(4,3),(5,5)\}$;
C) $R_3=\{(1,1),(1,3),(3,5),(3,7),(5,7),(7,3)\}$;
D) $R_4=\{(1,3),(2,5),(4,7),(5,9),(3,1)\}$;
I think the answer is both option A and D because $R_1 = \{(1,3),(3,5),(5,7)\}$. But when I see the solution it is given as options D. Am I missing something? Or is the answer provide wrong? Please elaborate so that I can understand.
 A: You're quite right in that
$$
R_1 = \{(x,y):y=x+2,\ x \in X,\ y \in Y\} = \{(1,3),(3,5),(5,7)\}
$$
which is a relation over $X$ and $Y$.  You can also tell it's a relation because the "$x \in X, y \in Y$" part of its definition implies $R_1 \subseteq X \times Y$.  I'm guessing the author intended the first set to be
$$
\{(x,x+2):x \in X\}
$$
which would not be a relation over $X$ and $Y$.  (Or maybe the author meant to say "function" instead of "relation", as in Theo Bendit's comment.)
We know $R_2$ is not a relation since $(1,2) \not\in X \times Y$.  Both $R_3$ and $R_4$ are relations, and you can check the first coordinates all belong to $X$ and second coordinates all belong to $Y$.
A: As Theo Bendt says in his comment, a relation between sets $X$ and $Y$ is simply a subset of $X \times Y$, that is, if you choose $R \subseteq X \times Y$, then you can define a relation, let's call it $\lt_R$, by stating that $x \lt_R y$ for $x \in X, y \in Y$ if and only if $(x,y) \in X \times Y$.
Thus, $R_1$ does not define a relation, since $R_1=\{(1,3),(2,4),(3,5),(4,6),(5,7)\}$, which is not a subset of $X \times Y$, as $4 \notin Y$.
Applying the same analysis shows that $R_2$ also doesn't define a relation, but $R_3$ and $R_4$ both do.
