Question

If we are given two sets as $Q=\\{2,6,9\\},R=\\{2,4,7\\}$, find $Q\times R$

Answer 1

We here have been given two sets Q and R and we have to find $Q\times R$. For this, we will first tell what is meant by $Q\times R$ which is that it is a set containing all the possible relations from Q to R. Then, we will find all those relations and keep them in a set. Hence, we will contain the answer.

Complete step-by-step solution
We here have been given two set Q and R defined as:
Q= {2,6,9}
R= {2,4,7}
Now, we know that $Q\times R$ is a set that contains all the possible relations formed from set Q to set R.
Thus, the value of $Q\times R$ will be given as a set containing all the relations which are possible from Q to R.
Now, all the elements from set Q can form a relationship with all the elements of set R.
Thus, all the relations from Q to R formed by the first element, i.e. 2 are:
$\left( 2,2 \right),\left( 2,4 \right),\left( 2,7 \right)$
Similarly, all the relations from Q to R formed by the second element, i.e. 6 are:
$\left( 6,2 \right),\left( 6,4 \right),\left( 6,7 \right)$
Similarly, all the relations from Q to R formed by the third element, i.e. 9 are:
$\left( 9,2 \right),\left( 9,4 \right),\left( 9,7 \right)$
Hence, the set $Q\times R$ is given as:
$Q\times R=\\{\left( 2,2 \right),\left( 2,4 \right),\left( 2,7 \right),\left( 6,2 \right),\left( 6,4 \right),\left( 6,7 \right),\left( 9,2 \right),\left( 9,4 \right),\left( 9,7 \right)\\}$

Note: We here can use the formula to confirm if we have all the elements of $Q\times R$ or not. This formula is given as:
Number of elements in $Q\times R$= $m\times n$
Where m= number of elements in set Q
n= number of elements in set R
Using this formula, we can check the actual number of elements in $Q\times R$ and see if they’re equal to the number of elements we found or not.
Here, $m\times n=9$ and the set we wrote as $Q\times R$ also has 9 elements. Hence, we have got all the elements.