Lecture Exam Sample Spring 2026

Discrete Mathematics - Lecture Exam Sample - Spring 2026

NAME:
NEPTUN code:
Group number/Class Teacher:

Exam details

• Duration: 120 minutes.
• Total score: 34 points.
• Success criterion: At least 9 marks from Part 1 and at least 7 marks from Part 2.
• Equipment: Only blank paper and pen – no notes or calculators.

Instructions

• Each task must be solved on paper using pen.
• Write your name, Neptun code, on the top of this test paper; also write your name and Neptun code on every page you use.
• After finishing, place all solution papers behind this test paper and fold them in half (parallel to the longer side) so they stay together.
• Mobile phones must be kept in bags throughout the exam.

Justification required

Justify your answers to proof questions — brief justifications suffice. A yes/no answer without justification scores 0 marks (except Part 1 short questions).

Leave it unevaluated

You may leave binomial coefficients or factorials as-is — no calculator is needed.

Info

In Part 1 short questions, when asked to write down a theorem/statement, no proof is required.

---

Questions

Part 1: Short questions (18 marks)

Proofs are not required – only the answers.

1. (1 mark)

Write down two properties of the inclusive or ($\vee$) logical operation learnt. (You can choose any two properties learnt of inclusive or covered in the course, except for the definition of inclusive or. Please write down each property you choose using a formula; writing the names of the properties is optional.)

---

2. (2 marks)

(a) Define the Cartesian product of two sets. (1 mark)

(b) Consider the sets $A=\{1,2,3\}$ and $B=\{a,b,c\}$. Find the Cartesian product of $A$ and $B$. (1 mark)

---

3. (1 mark)

Define what the composition of two binary relations is.

---

4. (2 marks)

(a) Define what it means for a binary relation $R\subseteq X\times X$ to be transitive. (1 mark)

(b) Give an example for a transitive binary relation on the set $A=\{a,b,c,d,e\}$. (1 mark)

---

5. (2 marks)

(a) Define what an equivalence relation is. (If you are referring to some properties of the relation in your answer, you do not necessarily need to write down the definitions of those properties, naming them is sufficient here.) (1 mark)

(b) Give an example for an equivalence relation $\sim$ on a (finite or infinite) set $X$. Please specify both the equivalence relation $\sim$ and the set $X$. (1 mark)

---

6. (2 marks)

(a) What does it mean for a function $f:X\to Y$ to be surjective? (1 mark)

(b) Consider the sets $A=\{a,b,c,d\}$ and $B=\{1,2,3\}$. Give an example for a surjective function $f:A\to B$. (1 mark)

---

7. (2 marks)

(a) Write down four properties of the absolute value and/or conjugate of complex numbers learnt. (You can choose any four properties learnt of the absolute value and/or conjugate, except for the definitions. You need to choose four properties in total.) (1 mark)

(b) Consider the complex number $z=-3-2i$. Find $|z|$ and $\bar{z}$. (1 mark)

---

8. (1 mark)

State the theorem about the number of permutations with repetition of a finite set. (Please write a complete sentence, referring to the meaning of any notation - i.e. symbols/letters - you are using in the formula.)

---

9. (2 marks)

(a) Formulate the binomial theorem. (Please write a complete sentence, referring to the meaning of any notation - i.e. symbols/letters - you are using in the formula.) (1 mark)

(b) Write down the binomial expansion of $(x+y{)}^6$. (1 mark)

---

10. (1 mark)

State two properties of the binomial coefficients learnt. (Please express the properties using formulas; naming the properties is optional.)

---

11. (1 mark)

State the Pigeonhole principle.

---

12. (1 mark)

Write down the definition of an undirected graph. (Please provide the formal definition learnt.)

---

Part 2: Proof questions (16 marks)

Please note that for a full mark you need to write down both the theorem/statement asked for and its proof. Scoring: theorem/statement: 1 mark; proof: 3 marks.

P1. (4 marks)

Write down and prove three properties of set union learnt. (You can choose any three properties learnt, except for the definitions of set union. Please write down each property you choose using a formula; writing the names of the properties is optional.)

---

P2. (4 marks)

Write down and prove the theorem about the associativity of the composition of binary relations. (Please state the theorem using a complete sentence, referring to the meaning of any notation - i.e. symbols/letters - you are using in the theorem.)

---

P3. (4 marks)

State and prove the theorem about the formula for the $n^{\text{th}}$ roots of complex numbers in polar form. (Please write a complete sentence when formulating the theorem, referring to the meaning of any notation - i.e. symbols/letters - you are using in the formula.)

---

P4. (4 marks)

Formulate and prove the theorem about the number of combinations with repetition. (Please state the theorem using a complete sentence, referring to the meaning of any notation - i.e. symbols/letters - you are using in the formula.)

---

Grade boundaries

Grade	Requirement
2 (pass)	At least 9 marks from Part 1 and at least 7 marks from Part 2
3	At least 9 marks from Part 1, at least 7 marks from Part 2, and total score of at least 20
4	At least 9 marks from Part 1, at least 7 marks from Part 2, and total score of at least 25
5	At least 9 marks from Part 1, at least 7 marks from Part 2, and total score of at least 29