PEI Math 521B common mistakes
PEI Math 521B · Understand the traps and how to avoid them.
When asked to prove that a sum or product is divisible by n, the candidate substitutes one or two sample values, observes the pattern holds, and treats the pattern as proof instead of setting up induction or a congruence argument modulo n.
The problem is stated in a way that invites exploration, and checking small cases is the natural first step; the candidate mistakes the confidence gained from consistent cases for a complete argument, and may never have internalized that an infinite family of integers requires induction or a modular argument.
What to do instead
Before writing, state the quantifier explicitly ('for all n ≥ 1'), then commit to one of two templates: strong/simple induction with a clearly labeled base case and inductive step, or a direct congruence computation mod n. Reserve numerical checks for discovering the pattern, and never present them as the proof itself.
In a gcd/lcm or linear Diophantine problem, the candidate applies Euclid's algorithm correctly but then reports the gcd as the answer to a Bezout question, or forgets to divide through by gcd(a,b) before declaring the Diophantine equation solvable, concluding 'no solutions' when there are infinitely many.
The mechanics of the Euclidean algorithm are rehearsed separately from the existence theory of ax + by = c, so the candidate treats the algorithm as the whole topic; the divisibility condition gcd(a,b) | c is easy to state but easy to skip under time pressure.
What to do instead
Adopt a fixed checklist for ax + by = c: compute d = gcd(a,b) first, test d | c, and only then back-substitute, scaling the particular solution by c/d. Write the general solution as (x0 + (b/d)t, y0 − (a/d)t) so the parameter appears explicitly.
When proving a statement about groups, rings, or fields, the candidate verifies closure and the existence of an identity but silently omits associativity, inverses, or the commutativity of addition in a ring, and treats the omitted axioms as 'obvious'.
Textbook examples (integers, matrices, modular arithmetic) satisfy the missing axioms automatically, so the candidate has never had to check them; under exam pressure the list of axioms collapses to the two most memorable ones.
What to do instead
For any 'is this a group/ring/field?' question, write the full axiom list as a numbered column and tick each one with a one-line justification or an explicit counterexample. Practice on a non-example (e.g., odd integers under addition, or nonzero integers under multiplication) so the omissions become visible.
The candidate computes the order of a subgroup or the size of a quotient using Lagrange's theorem in the wrong direction - concluding that a group of order 12 must have a subgroup of order 6, or that an element of order 4 exists because 4 divides 12.
Lagrange's theorem is remembered as a biconditional ('divisibility iff subgroup') because the converse holds for cyclic groups, and the candidate generalizes from that special case without noticing the hypothesis.
What to do instead
Memorize the exact statement with its one-way implication and pair it with the standard counterexample: A4 has order 12 and no subgroup of order 6. When a problem needs the converse, look for cyclicity or use Cauchy/Sylow, and state which theorem supplies existence.
In a Euclidean-geometry proof, the candidate draws a figure that assumes the conclusion - placing the point on the angle bisector, or drawing the 'altitude' perpendicular - and then reasons from the picture rather than from the given hypotheses.
Diagrams are the primary tool for generating insight in geometry, and a well-drawn figure makes the desired configuration look inevitable; the candidate does not separate what is given from what is to be shown.
What to do instead
Before drawing, list the givens and the goal in two columns. Draw the figure with the goal deliberately false (e.g., a visibly non-perpendicular line) and mark only given information; every subsequent claim must cite a definition, postulate, or previously proved theorem.
The candidate uses congruence criteria (SSA, AAA) that are not valid, or applies 'CPCTC' in reverse - asserting triangles are congruent because corresponding parts are equal, when the parts equality is what needs proof.
The abbreviations are learned as a list without the counterexamples, and CPCTC is memorized as a phrase rather than as the final step of a congruence proof; SSA looks like SAS on a sketch.
What to do instead
Keep a short reference card: valid are SSS, SAS, ASA, AAS, and HL (right triangles only); invalid are SSA and AAA, with the classic ambiguous SSA sketch as the counterexample. In proofs, write 'triangles congruent by ___' before ever writing CPCTC.
On a counting problem, the candidate multiplies choices together when the objects are indistinguishable, or adds counts of overlapping cases without applying inclusion–exclusion, producing an answer that double-counts arrangements.
The multiplication principle is the first tool learned and is over-applied; distinguishing 'arrangements of identical items' from 'assignments of distinct items' requires a modelling step that candidates skip when they recognize a familiar-looking problem.
What to do instead
Before computing, answer three questions in writing: are the objects distinct, does order matter, and is repetition allowed? Then choose among permutation, combination, and multiset formulas. For 'at least one' conditions, compute the complement; for unions, write out the inclusion–exclusion terms explicitly.
The candidate proves a statement by assuming it and deriving a true statement, then declares the original proved - the converse error - or uses a proof by contradiction but never uses the negated hypothesis, so the contradiction is spurious.
Algebraic manipulation runs in both directions fluently, and it is easy to lose track of which direction the implications are valid; the structure of a contradiction proof (assume ¬P, derive ⊥) is not kept in view while doing the algebra.
What to do instead
Write implications with explicit arrows and check each is reversible or note where it is not. In a contradiction proof, write 'Assume ¬P' at the top and refer back to it at the point where the contradiction arises; if the assumption was never used, the argument is invalid.
When a problem asks to 'prove or disprove', the candidate commits to a proof attempt and, after failing, writes a vague disproof, or produces a single example as a disproof of a universally quantified claim without checking the quantifier.
Exam convention rewards a decisive answer, and candidates fear leaving a blank; the quantifier structure of the statement is not parsed before choosing a strategy, and 'counterexample' is confused with 'example'.
What to do instead
First rewrite the statement in quantifier form (∀ or ∃). A universal claim is disproved by one counterexample only; an existential claim needs a construction. If a proof stalls, spend two minutes searching small cases for a counterexample before returning to the proof.
The candidate solves a problem correctly in scratch work but loses marks because the final write-up omits the base case, the justification for a division step (dividing by a quantity that could be zero), or the conclusion sentence tying the result back to the question asked.
Scratch work is where the mathematics happens, and copying it to the answer sheet feels like transcription; the candidate assumes the reader can reconstruct the omitted steps, and time pressure discourages a final read-through.
What to do instead
Adopt a fixed proof skeleton - hypothesis, body, conclusion - and a habit of checking every division by a variable for the zero case. Reserve the last three minutes to reread each solution asking 'does the last line answer the question asked?'