$\left(A\times B\right)^n=A^n\times B^n$?

Question

Is this property true:

For any set $A$ and $B$:

$$\left(A\times B\right)^n=A^n\times B^n?$$

Where $A\times B$ is the Cartesian product and $A^n=\underbrace{A\times A\times \cdots\times A}_{n\, \text{times}}$.

Answer 1

Not quite, but almost.

Note that $(A\times B)^n$ is a set of $n$-tuples of ordered pairs, whereas $A^n\times B^n$ is a set of ordered pairs made of $n$-tuples.

However, there is a very natural bijection between $(A\times B)^n$ and $A^n\times B^n$.

Answer 2

Well

$(A\times B)^n = \{ ((a_1,b_1),(a_2,b_2), \dots ,(a_n,b_n)) , \; a_i\in A, b_i \in B\}$\

and

$(A^n \times B^n) = \{\left((a_1,a_2,\dots, a_n),(b_1,b_2,\dots,b_n)\right), \; a_i\in A, b_i \in B \}$

So there is clearly a bijection ($a_i \to a_i , b_i \to b_i$), note that i gave appropriate names to the elements to automatically define the bijection