Specify the filter and prove its expressive limits
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 2 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.
Declare every universe and every allowed operation before reasoning about a complement or expression.
Give witnesses or counterexamples with their quantified domains. For impossibility, prove a property of every permitted expression.
Attempt these changed settings before opening the public solutions. A finite table supports a calculation, not a universal expressibility claim.
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A finite set U contains reviewers. A nonempty family F consists of nonempty subsets of U; each member is a review panel. A subset T of U is proposed. Define, using only membership, equality, quantifiers and logical connectives, (i) T meets every panel, and (ii) one reviewer belongs to T and every panel. Include quantifier domains. Does (i) imply (ii)? Give a smallest-cardinality U counterexample and justify minimality. Prove (ii) implies (i). Finally define D to contain exactly reviewers in T who belong to some panel but not to every panel. Give its membership predicate and verify D = (T ∩ ⋃F) ∖ ⋂F by an equivalence chain. Explain why the assumptions on F matter to the smallest counterexample.
Structure your argument and preserve revisions
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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.
A log-filter language has alphabet Σ={a,b}. Expressions start with ∅,{a},{b} and use binary union, concatenation and complement relative to Σ*. No repetition operator is allowed. For any language K define S(K)=K∩{a}*, its pure-a slice. Prove the slice commutes with union and concatenation, and turns complement in Σ* into complement in {a}*. Use the unary finite-or-cofinite closure theorem to prove every expressible slice is finite or cofinite in {a}*. Can an expression accept exactly all words containing b together with all pure-a words of even length, including ε? Prove your answer. If repetition star is added, construct that target. Finally consider the language of words with equally many a and b: does this slice invariant alone prove it expressible or inexpressible?
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
A finite set U contains reviewers. A nonempty family F consists of nonempty subsets of U; each member is a review panel. A subset T of U is proposed. Define, using only membership, equality, quantifiers and logical connectives, (i) T meets every panel, and (ii) one reviewer belongs to T and every panel. Include quantifier domains. Does (i) imply (ii)? Give a smallest-cardinality U counterexample and justify minimality. Prove (ii) implies (i). Finally define D to contain exactly reviewers in T who belong to some panel but not to every panel. Give its membership predicate and verify D = (T ∩ ⋃F) ∖ ⋂F by an equivalence chain. Explain why the assumptions on F matter to the smallest counterexample.
A log-filter language has alphabet Σ={a,b}. Expressions start with ∅,{a},{b} and use binary union, concatenation and complement relative to Σ*. No repetition operator is allowed. For any language K define S(K)=K∩{a}*, its pure-a slice. Prove the slice commutes with union and concatenation, and turns complement in Σ* into complement in {a}*. Use the unary finite-or-cofinite closure theorem to prove every expressible slice is finite or cofinite in {a}*. Can an expression accept exactly all words containing b together with all pure-a words of even length, including ε? Prove your answer. If repetition star is added, construct that target. Finally consider the language of words with equally many a and b: does this slice invariant alone prove it expressible or inexpressible?
