article · 2016

Three lectures on (un)countability for Math 131 A

Benjamin Antieau

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Abstract

While the rational numbers Q are dense in the real numbers R, it seems like there are many, many more real numbers than rational. For instance, Q feels discrete, while R feels continuous. More concretely, every decimal expansion of a rational number is eventually repeating and conversely. There is no such restriction for irrational numbers, and certainly “most ” decimal expansions are not repeating. The goal of these lectures is to make this idea precise and to prove in a rigorous way that there are more elements of R than there are of Q. References for this material can be found in [1, Chapter 1] and [3, Chapter 2]. Recall that if A and B are sets, a function f: A → B assigns to every element x of A a unique element f (x) of B. A function f is • injective if f (x) = f (y) implies x = y; • surjective if for every z ∈ B there exists x ∈ A such that f (x) = z; • bijective if it is injective and surjective. Note that an injective function is also called one-to-one, while a surjective function is also called onto. A set with only finitely many elements is called finite. A non-finite set is called infinite. Another

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APA 7

Antieau, B. (2016). Three lectures on (un)countability for Math 131 A.

MLA 9

Antieau, Benjamin. “Three lectures on (un)countability for Math 131 A.” 2016.

Chicago

Antieau, Benjamin. 2016. “Three lectures on (un)countability for Math 131 A.”

Harvard

Antieau, B. (2016) ‘Three lectures on (un)countability for Math 131 A’.

BibTeX

@article{antieau2016three,
  title = {{Three lectures on (un)countability for Math 131 A}},
  author = {Antieau, Benjamin},
  year = {2016}
}

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