MH1812: pass the exams, not just read the notes
Your complete guide to Nanyang Technological University's discrete mathematics course. See where the marks are, work real practice questions, and study with an AI tutor that knows MH1812.
Sia generates MH1812 practice questions, walks through number theory and propositional logic: arguments step by step, and quizzes you on the material the exam weights most heavily.
Worked example
A connected graph has vertices of degrees 2, 2, 3, 3, 4, 4. Does it have an Euler circuit, and how many edges does it have?
Apply the handshaking lemma: the degrees sum to 18, so the number of edges is 18/2 = 9.
Euler's theorem: a connected graph has an Euler circuit only if every vertex has even degree, so there is no circuit; with exactly two odd vertices, an Euler path exists between them.
Option C doubles the edge count by forgetting the lemma; options B and D ignore the odd-degree vertices.
The trap: Reading the degree sum as the edge count. Each edge contributes to two degrees, so the sum must be halved — and the parity of individual degrees, not the total, decides Euler circuits. classic slip!
One exam decides 50% of your grade. Summative assessment. This whole page is built around that.
Overview
What MH1812 is, and where it sits
MH1812 Discrete Mathematics is NTU's discrete mathematics course for first-year computer science and computer engineering students, taught by the Division of Mathematical Sciences. The OBTL+ document frames it as an introduction to number theory, logic, combinatorics and graph theory, with familiarity with formal analysis through simple problems as the key objective rather than depth in any one structure.
Nine intended learning outcomes span the course: congruence modulo an integer; formulating and manipulating logical statements; identifying valid arguments; proving results by direct, inductive, contradiction and contrapositive methods; counting; solving linear recurrence relations; proving set equality; manipulating relations and functions; and basic graph theory including Euler and Hamilton cycles.
Assessment is two mid-semester short-answer quizzes at 25% each and a short-answer final examination at 50%, with no make-up quizzes. The course is taught as a flipped classroom — pre-recorded lectures, class time for discussion and problem-based learning, tutorials for line-by-line solutions — and is 3 AU with 38 contact hours, offered in Semester 1.
Always treat your own course outline and the exam timetable as authoritative.
Difficulty & time commitment
Is MH1812 hard, and how much time does it take?
MH1812 is manageable if you keep a weekly rhythm and treat the back half as the main event. The pattern is consistent: it starts gently and steepens, and the heaviest assessment is the part that separates grades.
The difficulty curve and the assessment weighting point the same way: the back half is harder and worth more. Front-loading effort there is the highest-return decision in the course.
Is this course for you
Who tends to do well, and who tends to struggle
You will likely do well if
- You write proofs in full sentences and can say which technique you are using.
- You watch the pre-recorded lectures before class; the class assumes them.
- You enjoy puzzles about arrangements, graphs and logic.
- You prepare for the two quizzes as separate exams, because they are.
You may struggle if
- You prove statements by checking examples.
- You miss a quiz — there is no make-up and it is 25%.
- You leave graph theory and functions to the last two weeks and they are 20% of the exam's ILOs.
- You confuse a statement with its converse; the logic weeks exist to fix exactly that.
- Practise negating quantified statements until 'for all' and 'there exists' swap automatically.
- Solve every recurrence twice: by backtracking and by the characteristic equation.
- Prove set equality by double inclusion in writing, both directions labelled.
- Draw small graphs and check Euler and Hamilton conditions by hand before using theorems.
Syllabus
The 13 topics, week by week
The exam-weight marker on each topic shows where the marks concentrate. The amber topics carry the highest exam weight.
T1 · Number theory and propositional logic
Week 1; ILOs 1-2Euclidean division, modular arithmetic; propositions, truth tables, De Morgan's laws.
T2 · Propositional logic: arguments
Week 2; ILOs 2-3Equivalence laws, inference rules, valid and invalid arguments.
T3 · Predicate logic
Week 3; ILO 2Predicates, quantifiers, negating quantified statements, truth values.
T4 · Predicate logic and proof techniques
Week 4; ILOs 3-4Conditional quantification; direct proof and proof by induction.
T5 · Contradiction, contrapositive and counting
Week 5; ILOs 4-5Proof by contradiction and contrapositive; principle of counting, permutations, combinations.
T6 · Recurrences and sets
Week 6; ILOs 6-7Linear recurrence relations by backtracking and characteristic equation; sets, cardinality, power sets.
T7 · Set theory
Week 7; ILO 7Cartesian products and proof of set equality by double inclusion.
T8 · Relations
Week 8; ILO 8Reflexivity, symmetry, antisymmetry, transitivity.
T9 · Equivalence relations and partial orders
Week 9; ILO 8Matrix representation, composition, ternary relations.
T10 · Graphs: Euler paths and cycles
Week 10; ILO 9Graphs, subgraphs, directed graphs, Euler's theorem.
T11 · Functions
Week 11; ILO 8Injectivity, surjectivity, bijectivity, inverses, composition.
T12 · Functions: pigeonhole and countability
Week 12; ILO 8Floor and ceiling, pigeonhole principle, countable sets, Cantor's diagonal argument.
T13 · Graphs: structure
Week 13; ILO 9Complete and bipartite graphs, handshaking lemma, adjacency matrices, Hamilton cycles, isomorphism.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Mid-semester quiz 1: short-answer questions (ILOs 1-3: number theory and logic) | 25% | Short-answer mid-semester quiz on number theory and logic (ILOs 1-3). First half of semester. Continuous assessment; no make-up. |
| Mid-semester quiz 2: short-answer questions (ILOs 4-7: proofs, counting, recurrences, sets) | 25% | Short-answer mid-semester quiz on proofs, counting, recurrences and sets (ILOs 4-7). Second half of semester. Continuous assessment; no make-up. |
| Final examination: short-answer questions (all ILOs) | 50% | Short-answer final examination on all nine ILOs. Examination period. Summative assessment. |
- The three components sum to 100 and no examination hurdle is published. There are no make-up quizzes; absence needs an email to the instructor in advance and a Singapore-issued medical certificate to the administrator.
- Quiz 1 covers ILOs 1-3 (number theory and logic) and quiz 2 covers ILOs 4-7 (proofs, counting, recurrences, sets), each 25%. The 50% examination covers all nine ILOs, so relations, functions and graph theory are examined only once, in the final.
This is an exam-cram course. With the exams at 50% of the grade and the final examination: short-answer questions (all ilos) alone at 50%, your result is overwhelmingly decided by how well you perform under time pressure. Summative assessment.
Final exam timing: During the examination period. Confirm the exact date and venue on your exam timetable.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The course rewards consistency over cramming, and practice over re-reading. Here is the loop that works, then what to have nailed before each exam.
The weekly loop
Before the mid-semester checklist
- Decide congruence modulo n and compute with modular arithmetic.
- Build a truth table and identify an invalid argument.
- Negate a quantified statement correctly.
- Write a direct proof and an inductive proof.
Before the final heaviest topics
- Solve a linear recurrence by the characteristic equation.
- Prove two sets equal by double inclusion.
- Classify a relation and a function by their properties.
- Apply Euler's theorem and the handshaking lemma to a graph.
The mistakes that cost marks
Induction without the hypothesis. State what you assume for k before proving k + 1; markers look for it.
Converse confusion. 'If p then q' does not give you 'if q then p'. Half of the logic quiz is about this.
Counting order twice. Permutations count order; combinations do not. Decide which before computing.
Teaching team
Who teaches MH1812
The bios below are factual. We do not rate lecturers; any star ratings are submitted by students who have taken MH1812.
Teaching team as listed in the course materials reviewed. AskSia does not rate lecturers; star ratings are submitted by students who have taken MH1812.
Formula & concept sheet
The vocabulary and formulas you must own
- Congruence modulo n
- Two integers are congruent mod n when n divides their difference.
- Truth table
- A tabulation of a compound statement over all truth assignments.
- Quantifier
- 'For all' or 'there exists', the building blocks of predicate logic.
- Proof by contrapositive
- Proving 'not q implies not p' to establish 'p implies q'.
- Combination
- A selection without regard to order.
- Linear recurrence
- A sequence defined by a linear rule on previous terms, solved via the characteristic equation.
- Power set
- The set of all subsets of a set.
- Equivalence relation
- A relation that is reflexive, symmetric and transitive.
- Bijection
- A function that is both injective and surjective.
- Euler circuit
- A closed walk using every edge exactly once; exists when every vertex has even degree.
- Handshaking lemma
- The sum of vertex degrees equals twice the number of edges.
Set texts
The prescribed reading
The syllabus references map straight onto these.
Discrete Mathematics with Applications
.
Where it fits
Prerequisites, related courses & why it matters
No prerequisites. Mutually exclusive with CE1001, CZ1001 and MH1301. 3 AU; 38 contact hours; offered in Semester 1.
Your MH1812 study toolkit
Study the course with Sia, not just read about it
Each tool already knows MH1812: your syllabus, your texts, and where the marks are. Grouped by how you study, from first contact to exam week.
FAQ
Frequently asked questions
Is MH1812 hard?
It rates moderate. The ideas are introductory, but the entire grade sits in three short-answer sittings with no make-ups, and proofs are marked on the argument rather than the answer.
What is the assessment breakdown?
Two mid-semester short-answer quizzes at 25% each and a final examination at 50%, per the OBTL+ document.
How is it taught?
As a flipped classroom: pre-recorded online lectures, class time for in-depth discussion and problem-based learning, and tutorials for line-by-line solutions.
What are the prerequisites?
None. It is mutually exclusive with CE1001, CZ1001 and MH1301.
Which textbooks?
Epp, Discrete Mathematics with Applications (4th ed.) and Rosen, Discrete Mathematics and Its Applications (6th ed.).
Who wrote the course?
The OBTL+ document names Gary Greaves as course author.
Study MH1812 with Sia
Work through number theory, propositional logic: arguments, predicate logic and the rest of the course with a tutor that knows it and quizzes you on the topics the assessments weight most heavily.
Start studying with Sia
