Discrete math practice, topic by topic

What makes discrete math different

Up through calculus, most math questions have a procedure. Discrete math is the course where that stops being true. The question is no longer "what is the answer" but "is this argument valid" — and a correct answer reached by faulty reasoning earns nothing.

That shift is why students who were comfortable in calculus can struggle here, and why the failure modes are so consistent. Almost nobody loses points on the arithmetic. They lose points by proving the converse, by negating half a statement, by writing an induction that never uses its own hypothesis, or by labeling a contraposition as a contradiction. Those are learnable, checkable errors, and they are what this practice targets.

Who this practice is for

• Undergraduates in a first discrete mathematics or discrete structures course
• Computer science and cybersecurity majors taking discrete math as a degree requirement
• Students in a transition-to-proofs or mathematical foundations course
• Anyone who can follow a proof when reading one but freezes when asked to start one

Topics in the current release

Each topic can be selected independently, so you can drill the exact thing that is costing you points rather than reshuffling the whole course.

Choose a topic to read its explainer, then practice it in the free app.

Each row is a course unit with its number of practice questions; inside the app you can pick any of its 26 topics on its own.

What the bank covers

Every unit of a standard first course is now in the bank: logic and proofs, sets, functions and sequences (including characteristic functions and recurrence relations), algorithms and big-O, number theory, induction, counting (permutations, combinations, the binomial theorem, the pigeonhole principle and inclusion-exclusion), relations, and graphs and trees.

How the app handles proofs

An app cannot grade a written proof, and one that claimed to would be lying. So this one tests the part of proof writing that is objectively checkable — the structure.

1. Which line opens the proof. A proof by contraposition of p → q starts by assuming ¬q. A proof by contradiction starts by assuming both p and ¬q. Different sentences, and only one answers each question.
2. What the inductive hypothesis is, exactly. Not the claim for all n, not the claim at k+1 — the claim at k, written out.
3. Where a broken proof breaks. Real proofs with one bad step: a missing basis case, a division by a possibly-zero quantity, a converse proved by mistake.
4. Which counterexample kills a claim. Disproving a universal statement takes exactly one object that meets the hypothesis and fails the conclusion.

Alongside those, the bank uses typed numeric answers for anything with a definite value — how many rows a truth table has, how many of five statements are tautologies, the smallest n where an inequality first holds, how many basis cases a strong induction needs. Typing a number removes the one-in-four guess that multiple choice hands you.

Study guides for this subject

• Truth tables, connectives, and tautologies — start here: every connective, the one row where a conditional is false, and what a tautology actually is
• Converse, inverse, and contrapositive — which one is equivalent to the original, and the "only if" vocabulary that causes the swap
• How to negate a quantified statement — push the negation inward, flip every quantifier, and don't lose the conditional
• How to write an induction proof — the skeleton a grader wants, and the line where the hypothesis must appear
• Contraposition vs. contradiction — what each assumes, what each must produce, and how to spot a mislabeled proof
• Proof by cases and modular arithmetic — choosing an exhaustive case split, and how a modulus kills an equation
• Constructive existence proofs — exhibit a witness, then verify it; and why uniqueness is a separate claim
• Two ways to prove an equivalence — truth table versus a chain of named laws, and when each is what was asked
• Boolean arithmetic: proving logic laws with 0s and 1s — every connective as a 0/1 formula, and proving an equivalence by multiplying out
• Tournaments: every round robin has a Hamiltonian path — an induction on the number of players, with the insertion step case by case
• Congruences and Fermat’s little theorem: huge powers made small — the safe operations on congruences, huge powers made small, and why 341 breaks the converse
• Fibonacci and Zeckendorf’s theorem: a strong induction proof — the greedy representation and its strong induction proof
• Functions: injective, surjective and bijective — the first line of each proof, five examples, cardinality and the 2ⁿ subsets
• Orbits, orders and primitive roots — why x ↦ 3x mod 7 is a bijection, orbits, orders and primitive roots
• Permutations and arrangements: n! and P(m, n) — n! and P(m, n), the induction behind each, and counting functions

Frequently asked questions

Kestrel Exams Discrete Math material is independent study practice. Questions and guides are original and AI-generated, written against standard course topics.