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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.

Last updated date: 13th Jun 2024
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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.

Complete step-by-step answer:
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.