Describe $B$ and $|B|$

Question

Let $$A = \{ x \in \Bbb C : x^6 =1 \}\;\; \text{and}\;\; B=\{ x^3 : x \in A \}$$

Describe $B$ and $|B|$

Is it $\{1\}$ or roots of $x$?

Thank you

Answer 1

If you take the complex numbers ${\Bbb C}$ as underlying set, then $A$ is the set of 6th roots of unity: $$A = \{e^{2\pi i k/6}\mid k=0,\ldots,5\}.$$ Then $B$ is the set of 3rd powers of $A$, which are the 2nd roots of unity: $\pm 1$.

Answer 2

Sketch of solution: Start with describing $A$, explicitly. Which complex numbers are in $A$? Then take each element of $A$, and raise them to the third power. The collection of all the results is $B$. What complex numbers are those?