Proofs with predicates and syllogism

Question

Need some help with this question.

Prove that the following syllogism is valid by following the steps below.

My teapot is purple
My teapot holds water
--------------------
There exist purple things that hold water

a) Translate the syllogism into set-theoretic notation

        x∈A
       (x∈B)
      -------
      (A∩B≠∅) or (C∩B≠∅)??

b) Translate your set-theoretic notation into the notation of predicate logic

         x∈A
         x∈B
        -----
      ∃y∈Ay∈B ??

c) Give a proof that the syllogism is valid, similar to that given in the lecture 5 videos.

      S = A
      p(y)=y∈B
      p(x)∧(x∈S)⊨∃y∈Sp(y)
      S≠∅ ↔ ∃x∈U x∈S

  /** curious as to what this final question is actually looking for? **/

I would really appreciate any hints or feedback.

Answer 1

For (a), the conclusion should be $A \cap B \neq \varnothing $

For (b), I think you are expected to say:

$A(x)$

$B(x)$

--

$\exists y (A(y) \land B(y))$

For (c), (not sure what "lecture 5 videos" are, LOL) you have Conjunction introduction and Existential generalization. I think you are expected to write a proof tree of some sort. Maybe look in your textbook for examples, or follow the example here.

Answer 2

This answer will symbolize "my teapot" as $t$ which is the name of a particular object in the domain of "things". I will use $x$ as a variable for an unspecified object in this domain. I will symbolize "purple things" as $P$ and "things that hold water" as $H$.

Here is the syllogism:

My teapot is purple
My teapot holds water
--------------------
There exist purple things that hold water 

Here is a formulation in set-theoretic notation:

$t \in P$

$t \in H$

$P \cap H \neq \varnothing $

Here are two slightly different formulations in predicate logic:

$Pt$ or $P(t)$

$Ht$ or $H(t)$

$\exists x(Px \land Hx)$ or $\exists x(P(x) \land H(x))$

Using the notation of the first formulation, here is a proof presented in a Fitch-style proof checker to confirm its validity:

The inference rules used are conjunction introduction (∧I) and existential introduction (∃I).

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/