Number of elements in A x B

Number of elements of A × B = Number of Elements of A  ×  Number of elements of B

So, if set A has 3 elements, set B has 2 elements

Number of elements of A × B = 3 × 2 = 6

Let us check

If A = {1, 2, 3}, B = {3, 4}

A  × B = {(1, 3), (1, 4),

        (2, 3), (2, 4),

        (3, 3), (3, 4)}

Number of elements of A × B = 6

If A × A has 9 elements, then how many elements does set A have?

We know that

Number of elements of A × A = Number of Elements of A  ×  Number of elements of A

Number of elements of A × A = (Number of elements of A) ²

9 = (Number of elements of A) ²

(Number of elements of A) ² = 9

Number of elements of A = √9 = 3