Question

A relation R is defined from a set $A = \left\{ {2,3,4,5} \right\}$ to a set $B = \left\{ {3,6,7,10} \right\}$ as follows: $\left( {x,y} \right) \in R \Leftrightarrow x$ is relatively prime to $y$ . Express R as a set of ordered pairs and determine its domain and range.

Hint: Try to match the elements of the set A and B roughly using lines. Check for the pair of co-primes but the first number should be from set A and the second should be from set B. For domain take the set from where mapping starts and for range take the set where you map to.

Let’s first see what is the given information in the question. We are provided we two sets $A = \left\{ {2,3,4,5} \right\}$and $B = \left\{ {3,6,7,10} \right\}$ , we are asked to make a set R of ordered pair with the rule that first is relatively prime to second.
Before solving the problem, we need to know about ordered pairs. Here R is a set of ordered pairs, i.e. is set which contain pairs of like $\left( {x.y} \right)$ which follows a rule that$x$, from set A, is relatively prime or co-prime to$y$, which is from set B.
So basically, we need to find co-primes with the first number from set A and second number from B
Let’s start
Ordered pairs of co-primes with $x = 2$ : $\left( {2,3} \right),\left( {2,7} \right)$
Ordered pairs of co-primes with $x = 3:\left( {3,7} \right),\left( {3,10} \right)$
Ordered pair of co-primes with $x = 4:\left( {4,3} \right),\left( {4,7} \right)$
Ordered pair of co-primes with $x = 5:\left( {5,3} \right),\left( {5,6} \right),\left( {5,7} \right)$
Therefore, our set R of the ordered pairs is ready
$\Rightarrow R = \left\{ {\left( {2,3} \right),\left( {2,7} \right),\left( {3,7} \right),\left( {3,10} \right),\left( {4,3} \right),\left( {4,7} \right),\left( {5,3} \right),\left( {5,6} \right),\left( {5,7} \right)} \right\}$
The domain of an ordered pair set is defined as the set of values of the first number in the pair, i.e. $\Rightarrow Domain = A = \left\{ {2,3,4,5} \right\}$
And the range of an ordered pair set is defined as the set of values of second numbers in the pair, i.e. $\Rightarrow Range = B = \left\{ {3,6,7,10} \right\}$

Note: Write all the pairs in the set in a certain sequence or pattern, there is a huge possibility that you miss some pairs. Start from one end and check from another set in a sequence. Things can get complicated otherwise. An alternative approach to this problem is to start matching pairs from the second number, i.e. from set B to set A. But in that case, do not forget to write numbers from set A first and then the number from set B.