Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 120 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Specify the source, target and representation convention for each map.
Prove well-definedness, injectivity and surjectivity separately; equal cardinality does not make every proposed map a bijection.
Try the changed constructions before opening public solutions. A finite digit prefix or table is not evidence for an infinite-set theorem.
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Let I=[0,1). Use canonical binary expansions, excluding expansions eventually equal to1. Encode a triple (x,y,z)∈I³ by taking the first digit of x,y,z, then the second digit of each, and so on. Prove that the resulting number belongs to I and that this is an injection. Is it onto I? Test the canonical repeating stream100100100… and justify your answer using the separated streams. Construct an injection I→I³ and use the two-injection theorem to conclude a bijection exists. Explain precisely why that conclusion does not turn the original digit encoder into a surjection.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let A=B=N₀, with f(n)=2n and g(n)=2n+1. For the two-injection construction define A₀=A∖g(B), Aₖ₊₁=g(f(Aₖ)), and C=⋃ₖ≥0 Aₖ. Describe A₀,A₁,A₂ exactly, and compute membership in C for0 through9, justifying exclusions rather than checking only a finite number of levels. Define the resulting piecewise bijection h and compute h(0),…,h(9). Prove that its inverse is b↦b/2 when b∈f(C), and b↦2b+1 otherwise. Explain why g⁻¹ cannot be used on every n∈A.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
You are supplied a bijective enumeration q₀,q₁,… of Q and the distinct irrational sequence aₙ=n+√2. Construct a bijection H:R→R∖Q by sending qₙ to a₂ₙ, aₙ to a₂ₙ₊₁, and every remaining real to itself. Prove that these cases define a map, give its full inverse and prove both compositions are identities. Explain why fixing every irrational while moving rationals into the irrationals would fail. Which supplied data remove any need to choose a new countably infinite subset? Does this bijection claim that R and R∖Q have equal measure or identical order under H?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Let I=[0,1). Use canonical binary expansions, excluding expansions eventually equal to1. Encode a triple (x,y,z)∈I³ by taking the first digit of x,y,z, then the second digit of each, and so on. Prove that the resulting number belongs to I and that this is an injection. Is it onto I? Test the canonical repeating stream100100100… and justify your answer using the separated streams. Construct an injection I→I³ and use the two-injection theorem to conclude a bijection exists. Explain precisely why that conclusion does not turn the original digit encoder into a surjection.
Let A=B=N₀, with f(n)=2n and g(n)=2n+1. For the two-injection construction define A₀=A∖g(B), Aₖ₊₁=g(f(Aₖ)), and C=⋃ₖ≥0 Aₖ. Describe A₀,A₁,A₂ exactly, and compute membership in C for0 through9, justifying exclusions rather than checking only a finite number of levels. Define the resulting piecewise bijection h and compute h(0),…,h(9). Prove that its inverse is b↦b/2 when b∈f(C), and b↦2b+1 otherwise. Explain why g⁻¹ cannot be used on every n∈A.
You are supplied a bijective enumeration q₀,q₁,… of Q and the distinct irrational sequence aₙ=n+√2. Construct a bijection H:R→R∖Q by sending qₙ to a₂ₙ, aₙ to a₂ₙ₊₁, and every remaining real to itself. Prove that these cases define a map, give its full inverse and prove both compositions are identities. Explain why fixing every irrational while moving rationals into the irrationals would fail. Which supplied data remove any need to choose a new countably infinite subset? Does this bijection claim that R and R∖Q have equal measure or identical order under H?
