Question

Let $R = \\{ (a,b):a,b \in N\;and\;b = a + 5,a < 4\\}$. Find the domain and range of $R$.

Answer 1

A relation is defined in the question with some conditions for each term. In general, relations provide the connection among any two sets. By looking at the conditions we can identify all the values that would belong to this relation. Every relation is an ordered pair so there will be conditions given for each value in the pair.

Complete step by step solution:
The relation $R$ that is given to us is defined like this:
$R = \\{ (a,b):a,b \in N\;and\;b = a + 5,a < 4\\}$
According to the given conditions:
The values taken by the first term $a$ in the relation $(a,b)$ can only be less than $4$. But it is given that $a \in N$ (natural numbers), so $a$ takes the following values:
$\Rightarrow a = 1,\;2,\;3$
Then the values taken by $b$ should always be in the form of $a + 5$.
So all the values of $b$ will be as follows:
$\Rightarrow b = 1 + 5,\;2 + 5,\;3 + 5$
$\Rightarrow b = 6,\;7,\;8$
So finally the relation is made up of the following ordered pairs:
$R = \\{ (1,6),\;(2,7),\;(3,8)\\}$
After forming the relation we can find the domain and range.
The first value in the relation’s ordered pair forms the set called domain and the second value that is found within the relation’s ordered pair (that is found by the condition given with respect to the domain value) is said to be the range.
Then the first values in the ordered pair are: $1,\;2,\;3$
And second values in the ordered pair are: $6,\;7,\;8$

Therefore, domain and range of this relation can be written as:
Domain of $R = \\{ 1,\;2,\;3\\}$
Range of $R = \\{ 6,\;7,\;8\\}$.

Note:
Just like relations we have another concept called functions. Functions are a form of relations itself, but where every input has a specific output. Here the set containing all values with which the function is defined is the domain of a function $f(x)$. Then the set containing every value $f$ takes is the range of that function $f$.