DM Final exam sample

Page 1: Final Exam Instructions

• Faculty/School: Faculty of Engineering and Natural Sciences

• Department: Department of Mathematics and Natural Science

• Educational program: 6В

• Course title: MAT 125 Discrete Mathematics

• Important Note:

  • Final Exam will be invalid if:

    • a. Use a pencil, red, or green pen.

    • b. Fail to fill out the answer sheet.

    • c. Engage in cheating, including:

      • i. Communicating with other students or attempting to view another's exam.

      • ii. Referencing external materials (books, notes, electronic devices).

      • iii. Copying any exam content.

• Exam Regulations:

  • Students cannot leave the exam room during the test.

  • Duration: 100 minutes.

  • Cheating results in disqualification from the exam.

  • All answers must include solutions to be accepted.

  • Concerns about question accuracy can be reported using the appeal form on the next page.

• Instructor Details:

  • Points allocation mentioned for each answer.

Page 2: Appeal Form

• Process:

  • Fill in the question number and describe the issue. Provide your answer to the inconsistency noted.

  • There are multiple sections to fill out for distinct questions.

Page 3: Exam Questions

1. Set Membership Question:

   • Set: A = {a, b, c}

   • Examine true propositions regarding subsets and set inclusion. Options include subsets and their compositions.

2. Function Properties:

   • Given a function f with properties determined over specified sets X and Z.

3. Relation Types:

   • Analyze the relation R and its properties of reflexivity, symmetry, and transitivity.

4. Function Domains:

   • Characteristics determining the domain and co-domain of functions.

5. Pigeonhole Principle:

   • Calculate minimum enrollment in the context of multiple cities to ensure a minimum from one city.

6. Logic Equivalence:

   • Identify the equivalent forms related to logical implications.

Page 4: Graph and Functions

1. Graph Edges Calculation:

   • Calculate edges from the vertex degree data provided.

2. Poset Comparability:

   • Evaluate pairs of elements in a poset for comparability.

3. Boolean Functions:

   • Determine the dual of the given boolean function.

4. Bit String Count:

   • Calculate the total number of distinct bit strings of a specified length.

5. Propositions:

   • Identify a statement that does not qualify as a proposition.

6. Power Set Elements:

   • Determine the elements in the power set of a given set.

7. NOR Operation Representation:

   • Express boolean functions using the NOR operation.

Page 5: Boolean Function Properties and Set Theory

1. Properties of Boolean Functions:

   • Assess the validity of multiple false statements about boolean functions.

2. Universal Set and Union:

   • List elements of the union of given sets based on provided definitions.

3. Quantifier Statements:

   • Determine the truth value of various quantified statements regarding real numbers.

4. Inference Rules:

   • Explore rules for derivations and conclusions based on given premises.

Page 6: RSA and Proofs

1. RSA Key Generation:

   • Given primes, list possible values for e and find the unique d that satisfies conditions.

2. Proof of Irrationality:

   • Provide proof techniques, including contraposition and vacuous proof.

Page 7: Set Subsets

1. Odd and Even Subset Proportions:

   • Demonstrate that a nonempty set has equal amounts of odd and even-sized subsets.