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Mathematics · unit version 1

Infinite cardinality through explicit constructions

Control digit ambiguity, build a bijection from two injections, and absorb countably many elements with explicit domain and selection assumptions.

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Readiness and preparation

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What is needed to prove that a proposed map is a bijection?

Digit encoding and infinite rearrangement can hide missing or repeated outputs.

Compare reasoning and find a refresher

Check that every input has a valid unique output, prove injectivity, and show every target has a preimage. An explicit inverse can establish the last two properties.

Review maps and inverse reasoning

Where is the inverse of an injection defined?

The two-injection construction uses an inverse only on its valid domain.

Compare reasoning and find a refresher

On the image of the injection. It need not be defined on the entire named codomain.

Review actual domains and partial inverses

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Construct the map and verify its inverse

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Lesson sequence

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  1. Canonical digits and injections from a square

    Lesson details and objectives

    Resolve positional ambiguity, prove canonical expansions exist uniquely, and encode two real numbers without mistaking an injection for a bijection.

    • Prove existence and uniqueness of canonical base-b expansions on [0,1), including the exact ambiguity mechanism.
    • Construct and decode a canonical binary interleaving injection with its output domain justified.
    • Identify outputs outside the image and distinguish finite-prefix approximation from an infinite-sequence proof.

    6 practice tasks.

  2. Build a bijection from two injections

    Lesson details and objectives

    Construct the Cantor–Schroeder–Bernstein bijection and prove its domain, cross-branch injectivity and full coverage.

    • Define the forward-generated subset that chooses between two injective maps.
    • Prove the resulting piecewise map is well defined, injective and surjective, including empty cases.
    • Apply the construction to changed countable examples and an interval-square injection without assuming the desired bijection.

    6 practice tasks.

  3. Absorb countable additions with an explicit shift

    Lesson details and objectives

    Construct and invert a bijection using a supplied sequence, repair interval endpoints, and distinguish disjoint copies from ordinary unions.

    • Prove the countable absorption bijection by separating sequence, added-copy and fixed-point cases.
    • Construct finite shifts and explicit interval-endpoint bijections with complete inverse domains.
    • Identify tagging and supplied-sequence assumptions without silently invoking a choice principle.

    6 practice tasks.