find the relation $R$ and the domain of $R$?

Question

Let $A = \{3,4,5,6,7,8\}$ define a relation $R$ from $A$ to $A$ by,

$$R= \{(x,y) \,:\, y= x-1\}$$

what is the relation of $R$? and domain of $R$?

Answer 1

$R$ is the set of all ordered pairs of elements where each of which are elements of $A$ such that the first element is one larger than the second. If you want to write out all elements of $R$, that should be simple. Just write down all pairs of elements of $A$ such that the first element is one larger than the second.

Explicitly, $R$ would be $\{(4,3),(5,4),(6,5),(7,6),(8,7)\}$

Now... the domain is the set of all values that appear anywhere in the relation as the first element. The range is the set of all values that appear anywhere in the relation as the second element. In other words, the domain and the range are $\{4,5,6,7,8\}$ and $\{3,4,5,6,7\}$ respectively.

Note that $(3,2)$ is not an element of the relation because $2$ is not an element of $A$.