Set Theory and Predicate Calculus?

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Discussion Overview

The discussion revolves around a problem involving set theory and predicate calculus, specifically proving that an element belongs to a set based on given subset relationships. The context is primarily homework-related, as participants are preparing for a test and seeking assistance with the proof.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to begin the proof, indicating a lack of confidence in their understanding of predicate calculus.
  • Another participant suggests that the problem may require knowledge specific to a certain textbook.
  • A different participant offers to help and recommends using a visual aid, such as a Venn diagram, to model the relationships described in the problem.
  • A later reply provides a step-by-step reasoning process, concluding that if x(sub)1 is in P, then it can be shown that x(sub)1 must also be in R, while also noting the need to express this reasoning in predicate calculus.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants express varying levels of understanding and approaches to the problem. Some participants are unsure about the specifics of the textbook, while others provide differing methods for tackling the proof.

Contextual Notes

Participants have not fully resolved the mathematical steps required to express the reasoning in predicate calculus, and there may be dependencies on specific definitions or interpretations of the symbols used.

thename1000
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Set Theory and Predicate Calculus (12 points)
Given: P ⊆ Q
Q ⊆ (S ∩ T)
S ⊆ (R ∪ T^c)
x(sub)1 ∈ P
Use predicate calculus to prove x(sub)1 ∈ R.

Studying for a test but I don't have this worked out for me. I honestly don't even know where to start. I know what union, intersect, etc and all the symbols mean I'm just bad at the Predicate Calculus.
 
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I think you need help from someone who knows that particular textbook.
 
g_edgar said:
I think you need help from someone who knows that particular textbook.

Oh really its that specific? :( too bad
 
Nah, I might be able to help. I will look at it after class.

Why not start by drawing a picture (e.g., a Venn diagram) to see what a model of these sentences must look like? I find that pictures are especially helpful at suggesting proofs by contradiction.
 
thename1000 said:
Set Theory and Predicate Calculus (12 points)
Given: P ⊆ Q
Q ⊆ (S ∩ T)
S ⊆ (R ∪ T^c)
x(sub)1 ∈ P
Use predicate calculus to prove x(sub)1 ∈ R.

Studying for a test but I don't have this worked out for me. I honestly don't even know where to start. I know what union, intersect, etc and all the symbols mean I'm just bad at the Predicate Calculus.

If x_1\in P then, by the first line, x_1\in Q. By the second line x_1\in S and in T. By the third line then, x_1\in R or x_1\in T^c. But since x_1\in T, it can't be in T^c. Therefore x_1\in R.

Now all you have to do is express that in predicate calculus!
 

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