Suppose I am given a set $A = \{0,1,2,3,4,5,6,7,8,9\}$ and a relation $R$ on $A$ which is given by $x R y \iff y=2x+3.$
In order to write $R,$ how would I go answering this question? Would I just input all the values of the set $A$ as $x$ giving me $\{(0,3),(1,5),(2,7),(3,9),(4,11),(5,13),(6,15),(7,17),(8,19),(9,21)\}?$ Or perhaps do I only input numbers in $y=2x+3$ that would output numbers only between $0-9?$
Thanks in advance!
Let $A$ be the following set
$$A = \{0,1,2,3,4,5,6,7,8,9\}$$
and $R$ the relation on $A$ given by the following condition
$$xRy \iff y = 2x+3.$$
What you are trying to determine is the relation $R$ itself (by defining it in extension). I.e., the relation $R$ will be
$$R = \{(x,y) \in A \times A \mid y = 2x + 3\}.$$
Now, doing the arithmetic, you observe that
$$\begin{align} 3 = 2 \cdot 0 +3 && \implies (0,3) \in R\\ 5 = 2 \cdot 1 +3 && \implies (1,5) \in R\\ 7 = 2 \cdot 2 +3 && \implies (2,7) \in R\\ 9 = 2 \cdot 3 +3 && \implies (3,9) \in R \end{align}$$
and also that
$$\begin{align}\forall x \in A, x \geq 4 & \implies y \geq 11\\ & \implies y \notin A\\ & \implies (x,y) \notin R.\end{align}$$
This means that there is no more ordered pairs of $A \times A$ that belong to R.
Therefore
$$R = \{(0,3),(1,5),(2,7),(3,9)\}.$$