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Previous Year Question Papers

← Back to Mathematical Logic, Set Theory and Combinatorics

Question papers from previous sittings of this course, kept here for exam-pattern reference. Worked examples in the module notes tagged [Oct 2024] are drawn from or modelled on the paper below.

MAT1MN104 — Mathematical Logic, Set Theory and Combinatorics

End Semester Examination — October 2024

Programme: B.Sc. Mathematics Honours (CUFYUGP 2024) University: University of Calicut Semester: I | Credits: 4 Time: Two Hours | Maximum Marks: 70

Exam Pattern

Section Type Marks each Ceiling Strategy
A Short questions 3 24 Answer as many as you can
B Medium questions 6 36 Answer as many as you can
C Long questions 10 10 Answer any one of two

Total: 70 marks (24 + 36 + 10)

Section A

Answer any number of questions. Each carries 3 marks. Overall ceiling: 24 marks.

  1. Give truth tables for conjunction and disjunction of two propositions.

  2. Rewrite the proposition “for each integer x, there exists an integer y such that x + y = 0” symbolically.

  3. Define contradiction. Give example.

  4. Let A = {a, b, x, y, z}, B = {c, d, e, x, y, z}, and U = {a, b, c, d, e, w, x, y, z}. Find (A ∪ B)′ and A′ ∩ B′.

  5. Let A = 3, B = 5 and A ∩ B = 2. Find A ∪ B .

  6. List the elements of the Cartesian product A × B, where A = {1, 2} and B = {a, b, c}.

  7. Let A = $\begin{bmatrix} 2 & -3 & 7 \\ 0 & 1 & 1 \end{bmatrix}$ and B = $\begin{bmatrix} 1 & 5 & 0 \\ 2 & 0 & -1 \end{bmatrix}$. Find A + B.

  8. Find the number of ways of drawing a red queen or a black king from a standard deck of playing cards.

  9. Find the number of words that can be formed by scrambling the letters of the word SCRAMBLE.

  10. Suppose a card is drawn at random from a standard deck of playing cards. Find the probability that it will be a spade.

Section B

Answer any number of questions. Each carries 6 marks. Overall ceiling: 36 marks.

  1. Show that p → q ≡ ~q → ~p.

  2. Simplify the set expression (A ∩ B′) ∪ (A′ ∩ B) ∪ (A′ ∩ B′).

  3. Using the principle of inclusion–exclusion, find the number of elements in the union of three sets A, B, and C where A = 10, B = 15, C = 20, A ∩ B = 5, A ∩ C = 4, B ∩ C = 3, and A ∩ B ∩ C = 2.

  4. Define absolute value function and draw its graph.

  5. Find the number of positive integers ≤ 3000 and not divisible by 7 or 8.

  6. Let A = $\begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & -1 \end{bmatrix}$ and B = $\begin{bmatrix} 3 & -2 \\ 0 & 1 \\ -1 & 0 \end{bmatrix}$. Find AB and BA, if defined.

  7. Find the number of groups that can be formed from a group of seven marbles if each group must contain at least three marbles.

  8. Find the probability of obtaining at least one head when three coins are tossed.

Section C

Answer any one of the following. The question carries 10 marks. Maximum 10 marks.

  1. Let A = $\begin{bmatrix} 2 & -3 \\ 5 & 0 \end{bmatrix}$, B = $\begin{bmatrix} 1 & 0 & -1 \\ 2 & -3 & 5 \end{bmatrix}$ and C = $\begin{bmatrix} 0 & -2 & 1 \\ -3 & 0 & 4 \end{bmatrix}$.

    (a) Show that A + (−A) = O. (b) Show that A(B + C) = AB + AC.

    OR

  2. (a) Explain converse, inverse, and contrapositive of a proposition with examples. (b) Verify that ~(p ∨ q) ≡ ~p ∧ ~q and ~(p ∧ q) ≡ ~p ∨ ~q.

Notes on Notation (as used in this paper)

This paper confirms the notation used throughout Koshy’s textbook and, therefore, throughout this course’s notes:

  • Negation: ~p (tilde), not ¬p.
  • Set complement: A′ (prime), not Aᶜ.
  • Implication: p → q. Conjunction: ∧. Disjunction: ∨.
  • Cardinality: |A|.
  • Matrices written in bracket form, with entries in a 2×3 or 3×2 grid as in the book.
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