Determining the list elements of $U = \{(A,B)\in \mathcal P(X) ×\mathcal P(X)\mid A=(X−B)\}$

Question

Define $X = \{1,2,3,\ldots,n\}$, for some positive integer $n$. The set $U$, is defined as: $U =\{(A,B)\in \mathcal P(X) ×\mathcal P (X)\mid A=(X−B)\}$. If $n=3$, show the elements of $U$.

I started the problem by defining $X = \{1,2,3\}$. Then I tried to find $\mathcal P(X)$. It appears that $\mathcal P (X) = \{ \{\}, \{1\}, \{2\}, \{3\}, \{1,2\} ,\{1,3\}, \{2,3\}, \{1,2,3\}\}$.

I am stuck here. I don't know how to find the elements of $U$. Any help on this would be highly appreciated!

Update: The answer from the book is this: (I have no idea how to get there)

Answer 1

$$ A = X \setminus B $$ ist the same as $A\cap B= \emptyset$ and $A\cup B = X$

You only have to partition $\{1,2,3\}$ into two disjoints sets:

let $B$ run through all subsets and take $A=X\setminus B$

Answer 2

Just going by the definition you gave of $U$:

$$ U = \{(\varnothing,\{1,2,3\}), (\{1\}, \{2,3\}), (\{2\}, \{1,3\}), (\{3\}, \{1,2\}),(\{1,2,3\},\varnothing), (\{2,3\},\{1\}), (\{1,3\},\{2\}), (\{1,2\},\{3\})\} $$

Edit: Missed that the elements of $U$ were ordered pairs instead of sets, sorry.

Answer 3

Define $X = \{1,2,3,\ldots,n\}$, for some positive integer $n$. $U =\{(A,B)\in \mathcal P(X) ×\mathcal P (X) \mid A \cap B=\varnothing \}$.

$U_i= \{(A,B)\in \mathcal P(X) ×\mathcal P (X) \mid $Card(A)=i $ and $ Card(B)=n-i $ ,A=(X−B)\} $ then,

$U=\bigcup_{i=0}^{i=n} U_i $,

$|U|=|\bigcup_{i=0}^{i=n} U_i|$= $2^n$

$U_0= \{(\varnothing,X)\}$

$U_1= {(\{1\},\{2,3,\ldots n\}),(\{2\},\{1,3,\ldots n\}),\ldots,(\{n\},\{1,2,\ldots n-1\})} $

.

.

$U_n=\{(X,\varnothing)\}$.