Compose the rules and prove the reachability claim
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 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 the order of relational composition and check every required axiom.
For graph constructions, specify every pair orientation and prove the claimed paths or obstructions.
Attempt the changed cases before opening public solutions. Exhaustive finite checks are evidence about their finite domain, not substitutes for the general proof.
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On U={(0,0),(0,1),(1,0)}, define R by equal first coordinates and S by equal second coordinates. Use S∘R to mean first R, then S. Prove R and S are equivalence relations. Compare the two compositions on the ordered pair ((0,1),(1,0)) and prove S∘R is not symmetric. Now define E by existence of a finite chain of R-steps or S-steps, allowing a chain of length zero. Prove E is the smallest equivalence relation containing R∪S and identify E on this U. Repeat the two-composition comparison after adding the missing point (1,1).
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.
Partition a tournament into three nonempty blocks A,B,C. Orient every cross-block edge A→B, B→C and C→A. Within each block choose a linear ordering and orient every edge from earlier to later. Call the first vertex of a block its leader, and call a vertex a king if it reaches every other vertex in at most two edges. Prove the kings are exactly the three leaders. If the block sizes are a,b,c, compute each leader’s outdegree. For sizes2,3,1, identify all maximum-outdegree vertices and exhibit a king that does not have maximum outdegree. Explain why this does not contradict the theorem proved in the lesson.
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
On U={(0,0),(0,1),(1,0)}, define R by equal first coordinates and S by equal second coordinates. Use S∘R to mean first R, then S. Prove R and S are equivalence relations. Compare the two compositions on the ordered pair ((0,1),(1,0)) and prove S∘R is not symmetric. Now define E by existence of a finite chain of R-steps or S-steps, allowing a chain of length zero. Prove E is the smallest equivalence relation containing R∪S and identify E on this U. Repeat the two-composition comparison after adding the missing point (1,1).
Partition a tournament into three nonempty blocks A,B,C. Orient every cross-block edge A→B, B→C and C→A. Within each block choose a linear ordering and orient every edge from earlier to later. Call the first vertex of a block its leader, and call a vertex a king if it reaches every other vertex in at most two edges. Prove the kings are exactly the three leaders. If the block sizes are a,b,c, compute each leader’s outdegree. For sizes2,3,1, identify all maximum-outdegree vertices and exhibit a king that does not have maximum outdegree. Explain why this does not contradict the theorem proved in the lesson.
