Proving Numerical Equivalence of Real Number Intervals with S-B Theorem

In summary, the Schroeder-Bernstein Theorem states that if there are injections from two sets A and B into each other, then there exists a bijective correspondence between A and B. This can be applied to prove that any two intervals of real numbers are numerically equivalent, as there exists an injective (and therefore bijective) function between them when represented on a graph.
  • #1
nickmai123
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Homework Statement



Using the Schroeder-Bernstein Theorem, prove that any two intervals of real numbers are numerically equivalent.

Homework Equations



Schroeder-Bernstein Theorem: Let [tex]A[/tex] and [tex]B[/tex] be sets, and suppose that there are injections from A into B and B into A. Then, there exists a bijective correspondence between A and B.

The Attempt at a Solution


None. I'm stuck. Can anyone help me with where to go?
 
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  • #2
If and only if two intervals are "numerically equivalent", there exists a bijective correspondence between A and B.
 
  • #3
Picture one of the intervals in the x-axis of the plane and the other in the y-axis. Can't you see the graph of an injective (in fact bijective) function between them?
 

What is the S-B Theorem and how does it relate to numerical equivalence of real number intervals?

The S-B Theorem, also known as the Sandwich Theorem or Squeeze Theorem, is a fundamental theorem in calculus that states that if two functions, f(x) and g(x), are "squeezed" between a third function, h(x), at a particular point, and the limit of h(x) as x approaches that point is equal to a common value, then the limits of f(x) and g(x) at that point will also be equal. This theorem is commonly used to prove the numerical equivalence of real number intervals by showing that the upper and lower bounds of the intervals can be "squeezed" between a single value, thus proving that the intervals are equivalent.

What is the difference between numerical equivalence and geometric equivalence of real number intervals?

Numerical equivalence of real number intervals refers to the equality of the numerical values between two intervals, while geometric equivalence refers to the equality of the graphical representation of the intervals. While the S-B Theorem can be used to prove numerical equivalence, it cannot be used to prove geometric equivalence.

What are the key steps in using the S-B Theorem to prove numerical equivalence of real number intervals?

The key steps in using the S-B Theorem to prove numerical equivalence of real number intervals are as follows:1. Identify the upper and lower bounds of the two intervals.2. Find a third function that "squeezes" the upper and lower bounds between a single value.3. Show that the limit of the third function at that point is equal to a common value.4. Use the S-B Theorem to conclude that the limits of the two intervals are also equal.5. Therefore, the two intervals are numerically equivalent.

Can the S-B Theorem be used to prove the numerical equivalence of infinite real number intervals?

Yes, the S-B Theorem can be used to prove the numerical equivalence of infinite real number intervals. As long as the two intervals have the same upper and lower bounds, the theorem can be applied to show that the limits of the intervals are equal, thus proving their numerical equivalence.

Are there any limitations to using the S-B Theorem to prove numerical equivalence of real number intervals?

One limitation of using the S-B Theorem is that it can only be used to prove numerical equivalence, not geometric equivalence. Additionally, the theorem can only be applied when the two intervals have the same upper and lower bounds. If the intervals have different bounds, the S-B Theorem cannot be used and another method must be used to prove their equivalence.

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