How do I calculate $(A\times B)^2$ for $A=\{1,2\}$ and $B=\{a,b,c\}$?

Question

Let $A = \{1,2\}$ and $B = \{a,b,c\}$, find $(A \times B)^2$.

I found $(A \times B) = \{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$

But how do I find $$\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\} \times \{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\} $$ ? Is it $\{(1,a,1,a),(1,a,1,b),(1,a,1,c), \ldots\}$ ?

If I am wrong please show me the correct method.

Answer 1

You are wrong.

The set

$M:=\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\} \times \{(1,a),(1,b),(1,c),(2,a),(2,b), (2,c)\}$

consists of pairs of pairs.

For example $((1,a), (2,b)) \in M$.

Answer 2

You will get $36$ elements $$\{((1,a),(1,a)), ((1,a),(1,b)),..., ((2,c),(2,c))\}$$

Answer 3

For completeness sake, the result is :

$\{\\ ((1, a), (1, a)), ((1, a), (1, b)), ((1, a), (1, c)), ((1, a), (2, a)), ((1, a), (2, b)), ((1, a), (2, c)),\\ ((1, b), (1, a)), ((1, b), (1, b)), ((1, b), (1, c)), ((1, b), (2, a)), ((1, b), (2, b)), ((1, b), (2, c)),\\ ((1, c), (1, a)), ((1, c), (1, b)), ((1, c), (1, c)), ((1, c), (2, a)), ((1, c), (2, b)), ((1, c), (2, c)),\\ ((2, a), (1, a)), ((2, a), (1, b)), ((2, a), (1, c)), ((2, a), (2, a)), ((2, a), (2, b)), ((2, a), (2, c)),\\ ((2, b), (1, a)), ((2, b), (1, b)), ((2, b), (1, c)), ((2, b), (2, a)), ((2, b), (2, b)), ((2, b), (2, c)),\\ ((2, c), (1, a)), ((2, c), (1, b)), ((2, c), (1, c)), ((2, c), (2, a)), ((2, c), (2, b)), ((2, c), (2, c))\\ \}$