Proof Index
  • TOC
  • v2
  • Courses
  • Subreddit
  • Patreon
  • Blog
  • Search
    • ▾Bridge course
      • •Countable and uncountable sets
      • ▸Proof techniques
        • •Example of a nonconstructive proof
        • •Direct proof
        • •Disproof by counterexample
        • •Proof by cases
        • •Proving properties of absolute value
        • •Proof by contrapositive
        • •Proof by contradiction
        • •Challenge
        • •Proof by induction
        • •Proof by strong induction
        • •Proving de Moivre's theorem
        • •Visual proof
        • •Challenge
      • ▸Relations
        • •Equivalence relations
        • •Intersection of equivalence relations
        • •Representing relations using matrices
        • ▸Combining relations
          • •Composition of relations
        • •Inverse of composite relation
        • •Representing relations using digraphs
        • •Properties of relations
        • •Proving equivalence relations
        • •What is a relation?
        • •Finding the inverse of a relation on a finite set
      • ▸Functions (bridge course)
        • ▸Injective, surjective, and bijective
          • •Intro
          • •Is it bijective? (infinite domain)
          • •Is it injective?
          • ▸Is it surjective?
            • •Proof: Composite of surjections is surjection
          • •Proving properties of injective, surjective, and bijective functions
     › Bridge course

    Countable and uncountable sets

    1. Watch this video to learn how to prove sets equinumerous by establishing a bijection:

    25879Comparing the (unknown) sizes of sets
    Eddie Woo

    Also, read 14.1 Sets with Equal Cardinalities.

    2. Watch this video:

    25509Which set is bigger: the integers or the rational numbers?
    Eddie Woo

    3. Using what you know about the rational numbers, try to figure out why the Cartesian product of countably many sets must be countable. After you've thought about it, check out this Web page for an explanation.

    7906Cartesian Product of Countable Sets is Countable/Informal Proof
    ProofWiki

    4. Watch this video:

    27761Countable & Uncountable Infinities
    Eddie Woo

    Or this one:

    70152Episode 4: Uncountability of Real Numbers [#MathChops]
    Center of Math

    5. Watch this video:

    10My Favorite Set Theory Problem (just like a brain teaser)
    blackpenredpen

    6. Watch this video:

    11Cardinality Example with [0,1]
    MathDoctorBob

    Lessons and practice problems