Exploring Subsets: Understanding Integers and Their Relationships in Mathematics

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Discussion Overview

The discussion revolves around the nature of subsets in the context of the set of integers, specifically examining whether certain mathematical expressions, such as \(\frac{\pi}{\pi}\) and \((\sin(x))^2 + (\cos(x))^2\), can be considered subsets of the integers. The scope includes conceptual clarification and mathematical reasoning.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants question whether \(\frac{\pi}{\pi}\) or \((\sin(x))^2 + (\cos(x))^2\) can be classified as subsets of the integers.
  • One participant asserts that \(\frac{\pi}{\pi}\) evaluates to 1, which is a member of the integers, thus making the set \{1\} a subset of the integers.
  • Another participant clarifies that while 1 is a member of the integers, the construction of 1 involves numbers not part of the integers, raising a question about the relevance of this construction.
  • One participant argues that the construction process does not matter, emphasizing that the final result is what determines membership in the integers.

Areas of Agreement / Disagreement

Participants express differing views on the significance of how numbers are constructed in relation to their membership in the set of integers. There is no consensus on whether the construction process affects the classification of subsets.

Contextual Notes

The discussion highlights the ambiguity surrounding the definitions and implications of subsets in mathematics, particularly regarding the role of the construction of numbers in determining their membership in the set of integers.

cragar
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This may be a dumb question but let's say i have the set of integers [itex]\mathbb{z}[/itex]
can I say that [itex]\frac{\pi}{\pi}[/itex] or [itex](sin(x))^2+(cos(x))^2[/itex]
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 [itex]\mathbb{z}[/itex]
can I say that [itex]\frac{\pi}{\pi}[/itex] or [itex](sin(x))^2+(cos(x))^2[/itex]
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 [itex]\pi[/itex] will not be a natural number, but [itex]\pi/\pi[/itex] 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.
 

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