Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 90 minutes, as an unvalidated planning estimate. This collection contains 10 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.
Try the entire set before revealing worked solutions. You may use paper for calculations and diagrams. The suggested time is guidance; no timer limits access.
Show your reasoning for every task, including domain restrictions and checks in the original statement. Exact-value matches assess only the requested result.
Written arguments require review against the supplied rubric. Recording a response or marking a lesson complete does not certify mastery.
These are public practice assessments. First responses and solution exposure are retained; another attempt is practice, not a new independent examination.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
Let f(n)=floor((3n+2)/5), g(n)=floor((5n−2)/3), both Z→Z. Classify each map and compute f⁻¹({−1}) and g⁻¹({0}). A learner claims f∘g is the identity. Test and repair that claim by constructing a right inverse r from the fiber formula. Prove f∘r=id but r∘f≠id. Explain why left and right inverses impose different properties, then classify n↦n|n| with codomain Z versus its exact image.
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.
In the numeral-free natural arithmetic language, specify that n is squarefree and has exactly two distinct prime divisors. You may use defined primality as a hygienic macro. Prove the specification from prime factorization, contrast it with merely having two distinct prime divisors, and determine n=1,6,12,30. Negate your formula with quantifier scopes explicit. Explain why a search observing two prime divisors is sufficient to refute a prime-power claim but insufficient to certify your stronger claim.
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.
Consider: “For every integer n there exists an integer m such that m>2n.” Write its negation and prove the original claim. Then reverse the quantifiers and refute the new claim.
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 be a nonzero integer. Prove that a divides 5b+3c whenever a divides both b and c. State the role of the integer witnesses.
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.
Prove, for every integer n, that n² is even if and only if n is even. Give both directions and identify where a contrapositive is useful.
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.
Prove that the rational numbers strictly below 1 have no largest element. Your construction must remain rational, remain below 1, and exceed the proposed largest number.
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.
For a function f:X→Y and subsets A,B of Y, prove f⁻¹(A∖B)=f⁻¹(A)∖f⁻¹(B) using element membership. Explain why f⁻¹ here does not require an inverse function.
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 f:X→Y and g:Y→Z. Suppose g is injective and g∘f is surjective onto Z. Prove f is surjective onto Y. Show with explicit finite sets and maps why the injectivity hypothesis on g cannot simply be removed.
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.
Prove by induction that 1²+2²+…+n²=n(n+1)(2n+1)/6 for every positive integer n. Explain why checking the first hundred cases alone would not prove it.
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.
On the integers define a~b when 3 divides a−b. Prove this is an equivalence relation and describe its classes. A proposed replacement says a is related to b when a−b is positive. Identify two failed equivalence axioms and give counterexamples.
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 f(n)=floor((3n+2)/5), g(n)=floor((5n−2)/3), both Z→Z. Classify each map and compute f⁻¹({−1}) and g⁻¹({0}). A learner claims f∘g is the identity. Test and repair that claim by constructing a right inverse r from the fiber formula. Prove f∘r=id but r∘f≠id. Explain why left and right inverses impose different properties, then classify n↦n|n| with codomain Z versus its exact image.
In the numeral-free natural arithmetic language, specify that n is squarefree and has exactly two distinct prime divisors. You may use defined primality as a hygienic macro. Prove the specification from prime factorization, contrast it with merely having two distinct prime divisors, and determine n=1,6,12,30. Negate your formula with quantifier scopes explicit. Explain why a search observing two prime divisors is sufficient to refute a prime-power claim but insufficient to certify your stronger claim.
Consider: “For every integer n there exists an integer m such that m>2n.” Write its negation and prove the original claim. Then reverse the quantifiers and refute the new claim.
Prove that the rational numbers strictly below 1 have no largest element. Your construction must remain rational, remain below 1, and exceed the proposed largest number.
For a function f:X→Y and subsets A,B of Y, prove f⁻¹(A∖B)=f⁻¹(A)∖f⁻¹(B) using element membership. Explain why f⁻¹ here does not require an inverse function.
Let f:X→Y and g:Y→Z. Suppose g is injective and g∘f is surjective onto Z. Prove f is surjective onto Y. Show with explicit finite sets and maps why the injectivity hypothesis on g cannot simply be removed.
On the integers define a~b when 3 divides a−b. Prove this is an equivalence relation and describe its classes. A proposed replacement says a is related to b when a−b is positive. Identify two failed equivalence axioms and give counterexamples.
