Question

If G={7,8} and H={5,4,2}, find $(G \times H)$ and $(H \times G)$.

Answer 1

Hint: At first what is the concept of the cross of any two sets. Now here we have given two sets namely G and H so using the concept we have with us, we will find the sets $(G \times H)$ and $(H \times G)$
$(A \times B)$ is defined as a relation, $(A \to B)$ where the elements of $(A \times B)$ will be in the form (a,b) where
$a \in A$ and $b \in B$.
$A \times B = \{ (a,b):a \in A,b \in B\} $

Complete step by step solution: Given data: G={7,8}
H={5,4,2}
Now we know that if we have two sets let A and B then
$(A \times B)$is defined as a relation, $(A \to B)$ where the elements of $(A \times B)$ will be in the form (a,b) where
$a \in A$ and $b \in B$.
$A \times B = \{ (a,b):a \in A,b \in B\} $

Therefore we can say that
$(G \times H) = \{ (7,5),(7,4),(7,2),(8,5),(8,4),(8,2)\} $
And $(H \times G) = \{ (5,7),(5,8),(4,7),(4,8),(2,7),(2,8)\} $

Note: We know that if two sets let X and Y have m and n numbers of elements respectively then the number of elements in the sets $(X \times Y)$or $(Y \times X)$ will be the product of the number of elements in the respective sets i.e. mn.
We can also check from sets we obtained in the above solution
i.e. the number of elements of $(G \times H) = 2 \times 3 = 6$
and the number of elements of $(H \times G) = 3 \times 2 = 6$