Prove that $∀A ∈ \mathscr P(U)∃!B ∈ \mathscr P(U) ∀C ∈ \mathscr P(U) (C ∩ A = C \setminus B)$.

Question

This is Velleman's exercise 3.6.8.b (And of course not a duplicate of Uniqueness proof for $\forall A\in\mathcal{P}(U)\ \exists!B\in\mathcal{P}(U)\ \forall C\in\mathcal{P}(U)\ (C\setminus A=C\cap B)$):

$\mathscr P$ is used to denote the power set.

$∃!B$ means that "there exists a unique set B such that..."

And here's my proof of it:

Proof.

Existence. Let $B = (U\setminus A) ∈ \mathscr P(U)$. Then clearly for all $A ∈ \mathscr P(U)$ and $C ∈ \mathscr P(U)$, $C\setminus B = C\setminus (U\setminus A) = C ∩ A$.

Uniqueness. To see that $B$ is unique, we choose some $B'$ in $\mathscr P(U)$ such that for all $A ∈ \mathscr P(U)$ and $C ∈ \mathscr P(U)$, $C ∩ A = C \setminus B')$. Then in particular, taking $C = U$, we can conclude that $U ∩ A = U \setminus B'$ which is equivalent to $A = U \setminus B'$ and thus $B' = U\setminus A = B$.

Here is my question:

Is my proof valid? Particularly "$C\setminus (U\setminus A) = C ∩ A$" and "$A = U \setminus B'$ and thus $B' = U\setminus A = B$" parts. In other words, do the mentioned parts need more explanation\justification?

Thanks.

For uniqueness you could note that when $C=U$ we must have $A=U\cap A=C\cap A=C$ \ $B=U$ \ $B.$ — DanielWainfleet, Aug 25 '17 at 17:41

score 2 · Accepted Answer · answered Aug 25 '17 at 14:18

I dont's think that the passages that you mentioned need further details. However, there is a small errer in your proof. That's when you write “Uniqueness. To see that $B$ is unique, we choose some $B'$ in $\mathscr P(U)$ such that for all $A ∈ \mathscr P(U)$…”. The quantifier $\forall A\in\mathscr{A}$ comes first. Therefore, your proof should start with: “Uniqueness. Let $A\in\mathscr P(U)$. To see that $B$ is unique, we choose some $B'$ in $\mathscr P(U)$ for all $C\in\mathscr P(U)$…”

score 2 · Answer 2 · answered Aug 25 '17 at 14:31

I agree with José Carlos Santos and I noticed a similar error in the proof of existence when you write "Then clearly for all $A ∈\mathscr{P}(U)$ and...": again, the quantifier "for all $A \in \mathscr{P}(U)$" should come at the beginning of your proof. Indeed, you have already fixed an $A \in \mathscr{P}(U)$ just to define $B$, so saying "for all $A \in \mathscr{P}(U)$" after the definition of $B$ is an abstract nonsense. Therefore, you should write "Existence. Let $A \in \mathscr{P}(U)$ and $B=(U∖A)∈ \mathscr{P}(U)$. Then clearly for all $C∈\mathscr{P}(U)$...".

Prove that $∀A ∈ \mathscr P(U)∃!B ∈ \mathscr P(U) ∀C ∈ \mathscr P(U) (C ∩ A = C \setminus B)$.

2 Answers2