Cartesian Product for sets Recall that the Cartesian product $A\times A$ is defined as the set $\{(x,y) : x ∈ A \land y ∈ A\}$ . Thus if for
example $A = \{1,2,3\}, A\times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}$.
Consider a set $A$ is not empty where the number $|A|$ of elements of $A$ is 20 less than the number $|A\times A|$ elements in $A\times A$. Thus $|A| + 20 = |A\times A|$. Determine the number of elements in $A$.
My attempt: I thought from set $A$ there are only 3 elements, $\{1,2,3\}$? but and 9 elements in set $A\times A$ but then the math does not work because 3 + 20 does not equal 9.
Could someone help me please?
Thanks

The set you are asked to consider in the second part of your post is some arbitrary set $A$, not the example set which is used to clarify the definition of the Cartesian product.
So, let $A$ be a non-empty set with cardinality $n$ (we don't know what that cardinality is, so $n$ is effectively our variable). Then $$|A| + 20 = |A\times A| \iff n + 20 = n^2 \iff n^2 - n - 20 = 0\iff (n - 5)(n+4) = 0$$
Can you take it from here? Only one solution $n$ is greater than zero, so that would be your solutions for $|A|$.
