The image you uploaded contains questions from an exam in Discrete Mathematics from Lagos State University. Here’s a summary of the questions:

Question 1 (Mandatory)

• (a) A class contains 10 students (6 men, 4 women). Find the number of ways to:

  • (i) Select a 4-member committee.
  • (ii) Select a 4-member committee with 2 men and 2 women.
  • (iii) Elect a president and a treasurer.

• (b) Show with truth tables that the propositions:

  • $\neg (P \land Q)$ and $\neg P \lor \neg Q$ are logically equivalent.

• (c) Discuss the Law of Non-Contradiction and give three examples.

• (d) Mention two reasons for studying Discrete Mathematics.

• (e) Solve the system of equations using Matrix Method and Cramer's Rule:

  • (i) $\begin{aligned} 3x + 2y &= 3 \\ 2x + 3y &= 7 \end{aligned}$
  • (ii) $\begin{aligned} 2x + 3y + 2z &= -3 \\ x + y + z &= 0 \\ -x + 2y - 3z &= -1 \end{aligned}$

Question 2 (Choose any 2)

• (a) Given the set $A = \{a, b\}, B = \{c\}, C = \{d, e, f\}$:

  • (i) List the elements of $A$.
  • (ii) Find $n(A)$.
  • (iii) Find the power set of $A$.

• (b) Let $A = \{1, 2, 3\}$, $B = \{a, b, c\}$, and $C = \{x, y, z\}$, and consider the relations:

  • $R = \{(1, b), (2, a), (2, c)\}$ from $A$ to $B$.

  • $S = \{(a, y), (b, x), (c, y), (c, z)\}$ from $B$ to $C$.

  • (i) Find the composition relation $R \circ S$.

  • (ii) Determine the matrices $M_R$ and $M_S$ for the respective relations.

• (c) Given the universal set $N = \{1, 2, 3, \ldots\}$ and subsets:

  • $A = \{n \mid n \leq 6\}$,
  • $B = \{n \mid 4 \leq n \leq 9\}$,
  • $C = \{1, 3, 5, 7, 9\}$,
  • $D = \{2, 3, 5, 7, 8\}$:
  • Find $A \cap B$, $B \oplus C$, and $A \cap (B \oplus D)$.

• (d) Algorithm analysis: Examine the given pseudocode and explain its output.

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Related Questions

1. How do you calculate combinations when selecting members from a group?
2. What are De Morgan's Laws and how do they relate to logical equivalences?
3. How can Cramer's rule be used to solve systems of linear equations?
4. What is the composition of relations and how do we interpret it?
5. How is the power set of a set computed?

Tip

When solving systems of equations using matrices, remember that the determinant plays a critical role in Cramer's Rule. If the determinant is zero, the system either has no solution or infinitely many solutions.