Representations that preserve exactly the right information
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 5 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.
State the vertex order, edge direction and whether loops are included before using an adjacency matrix.
Prove both directions of a representation claim. Equal statistics alone are not an isomorphism certificate.
Work from the changed cases before consulting public solutions; written proofs need review against the rubric.
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On all integers define xRy when x divides y, meaning y=kx for some integer k. Prove R is a preorder but neither antisymmetric nor symmetric. Determine the mutual-divisibility classes including zero, and prove the quotient is isomorphic to nonnegative integers ordered by divisibility. Identify its least and greatest elements and show it is not a total order. Can the original R be both an equivalence relation and a partial order?
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.
Order D={1,2,3,6,9,18} by divisibility. Represent each d as 2^a3^b and prove this identifies D with the coordinatewise product {0,1}×{0,1,2}. List every cover, identify the minimal elements of S=D without 1 in its induced order, and decide whether S has a least element. Let B be the adjacency matrix of the upward cover graph and R that of the strict divisibility relation, both in order 1,2,3,6,9,18. Find (B³)_(1,18), (R²)_(1,18) and (R³)_(1,18) using walks rather than unexplained multiplication. Explain what changes when I is added to R and compute ((I+R)³)_(1,18).
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 directed graph G on 0,1,2,3 has exactly edges 0→1,0→2,1→3,2→3,3→0. Relabel by f(0)=c,f(1)=a,f(2)=d,f(3)=b. Construct the adjacency matrix of the relabeled graph H in order a,b,c,d and prove that relabeling carries length-k walks bijectively, including k=0. Use that correspondence to compute the length-two walk count from c to b and the length-three count from c to c. A proposed target instead replaces b→c with c→b. Give a specific edge witness refuting that proposed isomorphism, and explain why checking only a sorted total-degree list would be insufficient.
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.
Compare the simple undirected cycle C8 with the disjoint union of two copies of C4. Each has eight vertices, eight edges, all degrees two and no triangles. Give a nonisomorphism certificate despite those equal statistics. Now regard every undirected edge as two directed edges. Prove that the sum of all entries of the kth adjacency power is 8·2^k for both graphs for every k≥0, so that statistic still cannot distinguish them. Count four-edge closed walks from one vertex in each graph and use the sums of diagonal entries of the fourth powers as another distinguishing certificate.
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 any poset P define ↓x={z:z≤x}. Prove x≤y iff ↓x is contained in ↓y and deduce injectivity. Explain why the resulting map need not be onto all lower sets. In the coordinatewise order on {0,1}², show that L={(0,0),(1,0),(0,1)} is a lower set but is neither principal nor a Cartesian product of subsets of {0,1}. Prove ↓(p,q)=↓p×↓q for arbitrary product posets. Finally, show how L can be reconstructed as the union of the principal lower sets of its maximal elements; does that union become a single principal lower set?
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 all integers define xRy when x divides y, meaning y=kx for some integer k. Prove R is a preorder but neither antisymmetric nor symmetric. Determine the mutual-divisibility classes including zero, and prove the quotient is isomorphic to nonnegative integers ordered by divisibility. Identify its least and greatest elements and show it is not a total order. Can the original R be both an equivalence relation and a partial order?
Order D={1,2,3,6,9,18} by divisibility. Represent each d as 2^a3^b and prove this identifies D with the coordinatewise product {0,1}×{0,1,2}. List every cover, identify the minimal elements of S=D without 1 in its induced order, and decide whether S has a least element. Let B be the adjacency matrix of the upward cover graph and R that of the strict divisibility relation, both in order 1,2,3,6,9,18. Find (B³)_(1,18), (R²)_(1,18) and (R³)_(1,18) using walks rather than unexplained multiplication. Explain what changes when I is added to R and compute ((I+R)³)_(1,18).
A directed graph G on 0,1,2,3 has exactly edges 0→1,0→2,1→3,2→3,3→0. Relabel by f(0)=c,f(1)=a,f(2)=d,f(3)=b. Construct the adjacency matrix of the relabeled graph H in order a,b,c,d and prove that relabeling carries length-k walks bijectively, including k=0. Use that correspondence to compute the length-two walk count from c to b and the length-three count from c to c. A proposed target instead replaces b→c with c→b. Give a specific edge witness refuting that proposed isomorphism, and explain why checking only a sorted total-degree list would be insufficient.
Compare the simple undirected cycle C8 with the disjoint union of two copies of C4. Each has eight vertices, eight edges, all degrees two and no triangles. Give a nonisomorphism certificate despite those equal statistics. Now regard every undirected edge as two directed edges. Prove that the sum of all entries of the kth adjacency power is 8·2^k for both graphs for every k≥0, so that statistic still cannot distinguish them. Count four-edge closed walks from one vertex in each graph and use the sums of diagonal entries of the fourth powers as another distinguishing certificate.
In any poset P define ↓x={z:z≤x}. Prove x≤y iff ↓x is contained in ↓y and deduce injectivity. Explain why the resulting map need not be onto all lower sets. In the coordinatewise order on {0,1}², show that L={(0,0),(1,0),(0,1)} is a lower set but is neither principal nor a Cartesian product of subsets of {0,1}. Prove ↓(p,q)=↓p×↓q for arbitrary product posets. Finally, show how L can be reconstructed as the union of the principal lower sets of its maximal elements; does that union become a single principal lower set?
