27 topics
Discrete math help, from induction to graph theory
Discrete math is usually the first course where the answer is an argument rather than a number, and that switch is what catches people out. You can count the handshakes in a room of ten people and still have no idea how to write the lines that show it holds for n people. Nothing in calculus prepared you for being marked on whether your reasoning is airtight.
Where students get stuck
Induction: I write the base case and then I stall
The step almost everyone skips is using the assumption. Write the statement for n = k as a line you are now allowed to use, then write the statement for n = k + 1 and hunt for the k version hiding inside it. In a sum, split off the last term and what remains is exactly the k case, so replace it with what you assumed. That substitution is the proof; the algebra afterwards is bookkeeping. If you reach the end and never used the k case anywhere, you did not prove it by induction — you proved something else.
Permutations or combinations — under exam pressure I pick wrong
Ask one question of the problem: if I shuffle the chosen items, is that a different outcome. A four-digit passcode changes when you reorder it, so order counts and it is a permutation. A committee of four does not change, so it is a combination. Combinations are just permutations with the orderings divided back out, which is where the extra factorial in the denominator comes from. When you still cannot tell, count a tiny version by hand — choose two from four — and see which formula matches the list you wrote down.
The question gives me a recurrence relation and I don’t know where to start
Decide first whether you need one value or a closed form. For a value, just iterate: compute terms until you reach the one asked for and stop. For a closed form of a linear recurrence with constant coefficients, form the characteristic equation by replacing the n-th term with r to the n, solve it, then combine the roots with unknown constants. A repeated root picks up an extra factor of n. The initial conditions are used last, to pin the constants down — using them first is the mistake that makes the whole thing feel impossible.
Euler circuits and Hamiltonian circuits keep blurring together
Euler is about edges, Hamilton is about vertices. An Euler circuit walks every edge exactly once, and it exists precisely when the graph is connected and every vertex has even degree — a test you can run in seconds by counting degrees. A Hamiltonian circuit visits every vertex exactly once, and there is no such test; deciding it is genuinely hard, which is why textbooks only offer sufficient conditions like Dirac’s. So if a question seems to want a quick yes or no with a reason, it is almost certainly asking about Euler.
What's covered
Discrete Math topics you can work through with a tutor, generate practice on, or turn into flashcards and a study plan.
Logic and proof
- Propositional logic and truth tables
- Predicates and quantifiers
- Direct proof, contrapositive and contradiction
- Mathematical induction and strong induction
- Proof by cases and disproof by counterexample
Sets, functions and relations
- Set operations and Venn diagrams
- Power sets and Cartesian products
- Injective, surjective and bijective functions
- Equivalence relations and partitions
- Partial orders and Hasse diagrams
Counting
- Sum and product rules
- Permutations and combinations
- The binomial theorem and Pascal’s triangle
- Inclusion–exclusion
- The pigeonhole principle
- Discrete probability and expected value
Recurrences and algorithms
- Recurrence relations and closed forms
- Solving linear recurrences
- Big-O, big-Omega and big-Theta notation
- Loop invariants and algorithm correctness
- Number bases and modular arithmetic
Graphs and trees
- Graph terminology and representations
- Degree, paths and connectivity
- Euler and Hamilton circuits
- Trees, spanning trees and traversals
- Graph colouring
- Shortest path algorithms
Discrete Math questions
Is this the discrete math taken inside a computer science degree?
Yes, that is the usual home for it, and the topic groups above match a standard first or second year CS syllabus. It also covers the version taught in a mathematics department, which leans harder on relations, proof and combinatorics and lighter on the algorithm analysis.
Will it write my proof for me?
No. It will ask what you are trying to show, get you to state the assumption, and stop you at the line where the argument actually breaks. Proofs are graded on your reasoning, and a proof you did not construct tends to fall apart the moment the marker changes one condition.
Can I show it a proof I already wrote and find out what is wrong with it?
Yes. Type it out, share your screen, or hold the page up to the camera and it reads it. You get told which line the gap starts at and what a marker would object to — a missing base case, a case you never covered, an assumption used before it was justified.
Do the practice questions include proofs, or only calculation?
Both. You can ask for counting and recurrence drills that have numerical answers, or for proof prompts where the worked solution shows the full argument laid out line by line so you can compare it against the structure of your own attempt.
Stuck on discrete math right now?
Talk it through out loud, share your screen, and watch it worked out step by step on a whiteboard.
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