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Cardinality of Infinite Sets

In set theory, it has been shown that the set of irrational numbers has a cardinality greater than the set of natural numbers. That is, the set of irrational numbers is so large that it is uncountably infinite.

  1. Perform a search with the phrase, “Who first proved that the real numbers are uncountable?”
    1. Who first proved that the real numbers are uncountable?
    2. What was the significance of this proof to the development of set theory and by extension other fields of mathematics?
  2. Recent discoveries in the field of set theory include the solution to a 70-year-old problem previously thought to be unprovable. To learn more read this article:
    1. What does it mean for two infinite sets to have the same size?
    2. The real numbers are sometimes referred to as what?
    3. Summarize your understanding of the problem known as the “Continuum Hypothesis.”
    4. Malliaris and Shelah’s proof of this 70-year-old problem is opening up investigation in what two fields of mathematics?
  3. Summarize your understanding of infinity.
    1. Define what it means to be infinite.
    2. Explain the difference between countable and uncountable sets.
    3. Research the difference between a discrete set and a continuous set, then summarize your findings.

Set Notation

In arithmetic, the operation of addition is represented by the plus sign, +, but multiplication is represented in multiple ways, including ,×,,,×,, and parentheses, such as 5(3). Several set operations also are written in different forms based on the preferences of the mathematician and often their publisher.

  1. Search for “Set Complement” on the internet and list at least three ways to represent the complement of a set.
  2. Both the Set Challenge and Venn Diagram smartphone apps highlighted in the Tech Check sections have an operation for set difference. List at least two ways to represent set difference and provide a verbal description of how to calculate the difference between two sets AA and BB.
  3. When researching possible Venn diagram applications, the Greek letter delta, ΔΔ appeared as a symbol for a set operator. List at least one other symbol used for this same operation.
  4. Search for “List of possible set operations and their symbols.” Find and select two symbols that were not presented in this chapter.

The Real Number System

The set of real numbers and their properties are studied in elementary school today, but how did the number system evolve? The idea of natural numbers or counting numbers surfaced prior to written words, as evidenced by tally marks in cave writing. Create a timeline for significant contributions to the real number system.

  1. Use the following phrase to search online for information on the origins of the number zero: “History of the number zero.” Then, record significant dates for the invention and common use of the number zero on your timeline.
  2. Find out who is credited for discovering that the 22 is irrational and add this information to your timeline. Hint: Search for, “Who was the first to discover irrational numbers?”
  3. Research Georg Cantor’s contribution to the representation of real numbers as a continuum and add this to your timeline.
  4. Research Ernst Zermelo’s contribution to the real number system and add this to your timeline.
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