Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 85 minutes, as an unvalidated planning estimate. This collection contains 4 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.
Attempt these changed constructions before opening solutions. Define every recursive quantity and state the induction predicate.
Provide arguments for arbitrary finite objects, including every constructor and the empty or leaf case. Examples help test a claim but do not prove it.
Use the rubrics for critique and revision with a competent reader. This public practice records exposure; it is not an independently invigilated examination.
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Let F(x)=1/(x−2), E(y)=(y−1)/(y−1), and C(y)=0·(1/y), with strict expression evaluation. Determine exact domains and values of E∘F and C∘F. Compare them with the simplified constant expressions1 and0. Explain separately whether raw substitution already retains the F-domain guard. Then replace the outer expression by the literal constant0 and explain what changes.
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.
A finite ordered tree is built by Tip, Unary(T), or Fork(A,B,C). Let L,U,F count tips, unary nodes and ternary forks. Define these counts recursively and prove L=2F+1 for every such tree. Explain why unary nodes do not appear in the formula. A learner instead claims the total number N of nodes is 3F+1. Find a counterexample and repair the formula, proving it from the counts. Explain why a tree consisting only of a tip must be included.
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 finite ordered full binary trees define G(T,S) by replacing every leaf of T with a fresh copy of S: G(Leaf,S)=S and G(Node(A,B),S)=Node(G(A,S),G(B,S)). Define L as the number of leaves and M as recursive mirror. Prove, for all T and S, L(G(T,S))=L(T)L(S), and M(G(T,S))=G(M(T),M(S)). State the quantified induction predicates. Calculate leaf counts when T=Node(Leaf,Node(Leaf,Leaf)) and S=Node(Leaf,Leaf). Explain why swapping only the outer children is not the required mirror.
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 parenthesis word let b(k) be opening symbols minus closing symbols in its first k positions. Call a nonempty word primitive if b(length)=0 and b(k)>0 at every interior position 0<k<length. Prove that every word with nonnegative prefix balances and total zero has a unique decomposition into a sequence of primitive words, including the empty case. Show that primitive words are exactly (u) with u a balanced word. Decompose (()())()(()). Explain why uniqueness of this decomposition does not make the grammar S→SS | (S) | empty unambiguous.
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(x)=1/(x−2), E(y)=(y−1)/(y−1), and C(y)=0·(1/y), with strict expression evaluation. Determine exact domains and values of E∘F and C∘F. Compare them with the simplified constant expressions1 and0. Explain separately whether raw substitution already retains the F-domain guard. Then replace the outer expression by the literal constant0 and explain what changes.
A finite ordered tree is built by Tip, Unary(T), or Fork(A,B,C). Let L,U,F count tips, unary nodes and ternary forks. Define these counts recursively and prove L=2F+1 for every such tree. Explain why unary nodes do not appear in the formula. A learner instead claims the total number N of nodes is 3F+1. Find a counterexample and repair the formula, proving it from the counts. Explain why a tree consisting only of a tip must be included.
For finite ordered full binary trees define G(T,S) by replacing every leaf of T with a fresh copy of S: G(Leaf,S)=S and G(Node(A,B),S)=Node(G(A,S),G(B,S)). Define L as the number of leaves and M as recursive mirror. Prove, for all T and S, L(G(T,S))=L(T)L(S), and M(G(T,S))=G(M(T),M(S)). State the quantified induction predicates. Calculate leaf counts when T=Node(Leaf,Node(Leaf,Leaf)) and S=Node(Leaf,Leaf). Explain why swapping only the outer children is not the required mirror.
For a parenthesis word let b(k) be opening symbols minus closing symbols in its first k positions. Call a nonempty word primitive if b(length)=0 and b(k)>0 at every interior position 0<k<length. Prove that every word with nonnegative prefix balances and total zero has a unique decomposition into a sequence of primitive words, including the empty case. Show that primitive words are exactly (u) with u a balanced word. Decompose (()())()(()). Explain why uniqueness of this decomposition does not make the grammar S→SS | (S) | empty unambiguous.
