Multiplying sets

Question

Is there simpler way of doing this problem?

$$(\{1, 2, 3, 4, 5\} \times \{6, 7, 8, 9, 10\}) \cap (\{4, 6, 8, 9, 10\} \times\{2, 3, 5, 7, 11\})$$

I've been doing this problem and it's taking me a long time. How would I approach this problem? help me

Answer 1

Indeed there is a simpler way, and yay for asking about it!

Any pairs in both $(A \times B)$ and $(C \times D)$ must have a first element in both $A$ and $C$ as well as a second element in both $B$ and $D$.

Comparing ${1, 2, 3, 4, 5}$ and ${4, 6, 8, 9, 10}$, we see that they only have $4$ in common.

Comparing ${6, 7, 8, 9, 10}$ and ${2, 3, 5, 7, 11}$, we see that they only have $7$ in common.

So answer is $\{(4, 7)\}$.

In general (saying what I said above algebraically):

$$(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)$$