Proofs and Discrete Structures: Mistake Clinic
Use this after a first attempt. If no error occurred, critique a deliberate fault; do not fabricate personal mistakes to satisfy a quota.
Module-Specific Mistake Radar
| Mistake | Small revealing case | Repair evidence |
|---|---|---|
| Converse used for free | 2 is even but not divisible by 4 | Write and test both implications |
| Witness depends on a later variable | A proposed integer larger than every integer | Rewrite quantifiers and challenge the witness |
| Undefined expression treated as a predicate | 1/x at x=0 | Restrict the domain or define the missing outcome |
| Codomain replaced by image | Doubling Z into Z | Exhibit a missing odd output |
| Canonical representation said to forbid quotient functions | Reduced numerator of a rational | Prove canonical choice is unique; define the quotient map |
| Antisymmetric confused with asymmetric | Singleton diagonal | Write both definitions; self-pairs distinguish them |
| Closure notation hides missing self-pairs | Empty relation on {0} | Distinguish positive from zero-length reachability |
| Minimal mistaken for least | Divisibility on {2,3,6} | Show the two incomparable minima |
| Induction jumps over initial values | A step using n−4 | Derive which starting values the step cannot cover |
| Decrease mistaken for termination | Positive reals repeatedly halved | Supply a well-founded measure |
| Structural induction confused with unique representation | A set built in different insertion orders | Distinguish induction on derivations from defining a representation-independent function |
Practice Mistake Checks
For logic practice, check the whole negation and each hypothesis. For set/function/relation practice, check carrier membership, both inclusions and every witness. For induction practice, check base reachability, hypothesis scope and termination separately. For code katas, check invalid input, empty cases and representation assumptions.
Repair record
Record the original claim, smallest revealing example, violated assumption, corrected proof/code and a fresh transfer attempt. Keep assistance visible. Do not classify a valid proof as wrong just because it uses size induction, a single iff chain, overlapping exhaustive cases, or a different valid counterexample.