Exploring Subsets: Understanding Integers and Their Relationships in Mathematics

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The discussion centers on whether certain mathematical expressions, specifically \(\frac{\pi}{\pi}\) and \((\sin(x))^2 + (\cos(x))^2\), can be considered subsets of the set of integers \(\mathbb{Z}\). It is clarified that \(\frac{\pi}{\pi}\) evaluates to 1, which is an integer, thus making the set {1} a subset of \(\mathbb{Z}\). However, the construction of the number 1 involves non-integer components, prompting a debate on the relevance of the construction process. Ultimately, the consensus is that the final result is what determines membership in the integers, not the method of construction. The discussion concludes with an affirmation of the importance of the final answer in determining subset relationships.
cragar
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This may be a dumb question but let's say i have the set of integers \mathbb{z}
can I say that \frac{\pi}{\pi} or (sin(x))^2+(cos(x))^2
is a subset of the integers?
 
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cragar said:
This may be a dumb question but let's say i have the set of integers \mathbb{z}
can I say that \frac{\pi}{\pi} or (sin(x))^2+(cos(x))^2
is a subset of the integers?

The expression you gave (if I read the latex right) will always be 1. The set {1} will be a subset of Z if that is what you are asking.
 
No, 1 is member of the integers. {1} is a subset.
 
yes but the way we constructed 1 used numbers that are not part of the integers.
I don't know if that matters
 
cragar said:
yes but the way we constructed 1 used numbers that are not part of the integers.
I don't know if that matters

The way things are constructed doesn't matter. It's the final answer that will matter. So \pi will not be a natural number, but \pi/\pi will be, since the expression evaluates to 1, and that is all what matters.
 
ok thanks for your answer. I just wanted to be clear on it.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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