Step by step
The question
Question: find A × B, when A is {1, 2} and B is {3, 4, 5}.
A = {1, 2}, B = {3, 4, 5}
A × B means: every pair with the first number from A and the second from B.
A pair is written in round brackets.
A × B = every pair (first from A, second from B)
1 Make the table
Across the top: the elements of B.
Down the side: the elements of A.
Each cell is a pair: its row first, its column second.
1 with 3 is (1, 3).
2 List the pairs
We write the pairs row by row, inside curly brackets: (1, 3), (1, 4) and so on.
A × B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
Careful!
The order matters.
In A × B the first number always comes from A.
So A × B is not the same as B × A.
Careful: not (3, 1) in A × B ✗
3 Check
Let's check the count: 6 pairs, and 2 times 3 is 6.
No pair twice.
So it is right!
6 pairs = n(A) × n(B) = 2 × 3
Let's look back.
First: make the table.
Then: list the pairs.
Last: check.
Answer
Careful: not (3, 1) in A × B ✗