Intervals and their subsets proof

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SUMMARY

The discussion centers on the proof regarding intervals and their subsets, specifically addressing the statement that if I is an interval and A is a subset of I, then A can be classified as either an interval, a set of discrete points, or a union of the two. The participant illustrates a counterexample using the interval (-1, 1) and the subset A consisting of all rational numbers within that interval, demonstrating that A does not fit the proposed classifications. The conclusion drawn is that the initial assertion is incorrect unless the definition of union is broadened to include arbitrary unions of sets.

PREREQUISITES
  • Understanding of real number intervals
  • Familiarity with set theory concepts
  • Knowledge of rational and irrational numbers
  • Basic proof techniques in mathematics
NEXT STEPS
  • Study the properties of intervals in real analysis
  • Explore set theory, focusing on unions and intersections
  • Learn about the distinctions between rational and irrational numbers
  • Review proof techniques, particularly counterexamples in mathematical logic
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Students studying real analysis, mathematicians interested in set theory, and anyone engaged in mathematical proofs and logic.

hlin818
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Homework Statement


I reduced another problem to the following problem:

If I is an interval and A is a subset of I, then A is either an interval, a set of discreet points, a union of the two.

Homework Equations


The Attempt at a Solution



Is this trivial?
 
Last edited:
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It's false. For example, take the set (-1,1) and inside of it the set A of all rational numbers in between -1 and 1. Unless by union you mean any arbitrary amount of sets being unioned together, in which case it's a silly question because any set A is the union of the sets each containing a single point of A
 
Ah completely overlooked that, thanks. I'll post up the full problem because now I'm sort of stuck.
 

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