Can I prove set propositions using first-order logic?

I'm studying logic and sets and I have to say there's a strong similarity between the two. Most boolean/logic axioms also apply to sets. At the end of my course I also studied first-order logic (or predicate logic) and how one can actually define statements using first-order logic. This was quite a revelation to me because it is quite easy to prove first-order logic statements compared to set propositions (my brain just works better at decomposing first-order logic propositions). So I'm wondering, is it possible to prove set propositions using first-order logic? Here's an example of a set proposition I have to prove:

if C ⊆ B then A ⇒ C ⊆ A ⇒ B

(not that A ⇒ B is assumed to mean ¬A ∪ B)

Now I was thinking I could convert this into a first-order logic statement, such as:

Subset(C, B) ⇒ Subset(A ⇒ C, A ⇒ B)

As you can see however I can't seem to understand how to prove that in first-order logic. Anyway, perhaps I am confusing the two and this is not possible but I was wondering what you guys thought about proving these set statements using first-order logic.

• Do you use "$A\Rightarrow C$" as notation for some set? That is not standard. Which set does that mean? – Henning Makholm Jun 26 '15 at 13:21
• Sorry it means ¬A u C, will edit. – Luca Matteis Jun 26 '15 at 13:21
• You can do this, but your primitive boolean expressions should be: "if $x\in C\Rightarrow x\in B$, then ... " (In other words, your expressions are all of the form $x\in A$, $x\in B$, $x\in C$). – Michael Burr Jun 26 '15 at 13:26

You can translate between set algebra and logic, but you need to speak about elements in order to do so.

If you take a set algebra expression, you can rewrite it into set-builder notation from the bottom up as follows:

• A named set $A$ is the same as $\{x\mid x\in A\}$.

• For a union, first rewrite the operands until they have set-builder form, and then use $$\{x\mid \phi(x) \} \cup \{x\mid \psi(x) \} = \{x\mid \phi(x)\lor\psi(x)\}$$ where $\phi(x)$ and $\psi(x)$ are some formulas that have $x$ free.

• Similarly, intersections and complement correspond to conjunction and negation: $$\{x\mid \phi(x) \} \cap \{x\mid \psi(x)\} = \{x\mid \phi(x)\land\psi(x) \} \\ \{x\mid \phi(x) \}^\complement = \{x \mid \neg \phi(x) \} \\ \{x\mid \phi(x) \} \setminus \{x\mid\psi(x)\} = \{x\mid \phi(x)\land\neg\psi(x)\}$$

• When you have a relation between sets, you get plain logical formulas back: $$\{x\mid \phi(x) \} \subseteq \{x\mid \psi(x)\} \iff \forall x (\phi(x)\rightarrow \psi(x)) \\ \{x\mid \phi(x) \} = \{x\mid\psi(x)\} \iff \forall x(\phi(x)\leftrightarrow \psi(x))$$

It seems you already have this kind of translation in mind when you're using "$\Rightarrow$" and "$\neg$" as set algebra operations -- these symbols are usually used only as logical connectives. (Conversely, there is no conventional logical connective that by itself represents set difference).

The underlying similarity here is that the universe of sets with $\cup$, $\cap$, $\complement$, or the universe of formulas with $\lor$, $\land$, $\neg$ both constitute Boolean algebras. The $\{x\mid\cdots\}$ construction is a homomorphism between the two Boolean algebras, and the work mostly just consists of changing notation to that used on the other world.

• Very cool thanks. Going back specifically to my example, do you think first-order logic can be used to prove it? – Luca Matteis Jun 26 '15 at 13:40
• @LucaMatteis: Certainly you can. Your example becomes the inference $$\forall x(x\in C\Rightarrow x\in B) \vdash \forall x(x\in A\Rightarrow x\in C) \Rightarrow \forall x(x\in A\Rightarrow x\in B)$$ which you can then prove with your favorite proof formalism for first-order logic. – Henning Makholm Jun 26 '15 at 13:45
• Nice of course! By the way, what does ⊢ mean? I suppose ⇒, right? – Luca Matteis Jun 26 '15 at 13:53
• @LucaMatteis: $\vdash$ is the entailment (or provability or consequence) relation. $P\vdash Q$ means that the formula $Q$ can be proved if we're allowed to assume $P$. It is a property of first-order logic (and most other logics) that $P\vdash Q$ holds exactly when $\vdash (P\Rightarrow Q)$. But notice that $\Rightarrow$ combines formulas into a different formula (which may well be true in some concrete situation without being provable), whereas something that involves $\vdash$ is not a formula but an assertion outside logic about the existence of certain proofs. – Henning Makholm Jun 26 '15 at 13:59
• @Luca: That's a good question that I think deserves an answer that is longer than what fits in a comment. You should consider asking it as a question of its own. – Henning Makholm Jun 27 '15 at 13:02

You can convert many set theoretic statements to Boolean expressions. The most important thing to note is that the set $A$ is not appropriate for a Boolean expression because it's not a statement. You should replace $A$ with $x\in A$ because $x\in A$ can be true or false. The translations are as follows:

$$\begin{array}{c|c} \text{Set}&\text{Boolean}\\\hline A\subseteq B&x\in A\Rightarrow x\in B\\ A^c&\neg(x\in A)\\ A\cap B&(x\in A)\wedge (x\in B)\\ A\cup B&(x\in A)\vee (x\in B)\\ A\setminus B&(x\in A)\wedge (x\not\in B) \end{array}$$ From this list, you can derive most statements in basic set theory (but some get very complicated to compute).

Notes: $A^c$ is the complement of $A$ in some universe. $\neg(x\in A)\equiv x\not\in A$.

• Indeed I was already aware of this. My question was about how to apply this to the set proposition shown in my question. And then how to construct first-order logic statements given the proposition at hand. – Luca Matteis Jun 26 '15 at 13:35
• The statements in the problem don't really make sense because $A$ and $B$ are sets, so you shouldn't use Boolean expressions with them. It would be better to not mix the two notations (even though they are equivalent). – Michael Burr Jun 26 '15 at 13:38
• Actually, as Henning showed above, it can be done! Thanks anyway – Luca Matteis Jun 26 '15 at 14:48