Power set of given set $R$

Question

Let $A=\{1\}, B=\{2,3\}$

$A\times B=R = \{(1,2),(1,3)\}$

then $P(R) = \{ \emptyset,\{(1,2)\},\{(1,3)\}, \{(1,2),(1,3)\}\}$

Isn't this the way it should be?

or in this way

$P(R)$ = {{},{(1},{2)},{(1},{3)},{(1,2)},{(1,(1},{(1,3)},{2),(1},{2),3)},{(1,3)},{(1,2),(1},{(1,2),3)},{(1,(1,3)},{2),(1,3)},{(1,2),(1,3)}}

Answer 1

Your first solution is correct.

Note that your set $R$ has 2=n elements. So the powerset $\mathcal{P}(R)$ has $2^n=2^2=4$ elements.

The elements of $R$ are namely $(1,2)$ and $(1,3)$. So the elements of $R$ are ordered pairs.

Answer 2

To eliminate the second option you can this:

Let $A$ be a set with cardinality n. I.e. the number of elements a set contains. Let's denote cardinality of $A$ by $|A|$. Then $|P(A)|=2^n$, where $P(A)$ is the power set of $A$. Since $|R|=2$, then $|P(R)|=2^2=4$. You can conclude that the second option is not correct since its cardinality is not equal to 4.