Introduction to Discrete Structure &Linear Algebra (AL-401) - Important Questions

Explain types of relations including reflexive, symmetric, transitive and equivalence relations with suitable examples.

Write a short note on Hasse diagram and Lattice.

Using Venn diagrams, prove De Morgan's laws (A union B)' = A' intersection B' and (A intersection B)' = A' union B'.

Solve the second-order linear homogeneous recurrence relation a_n = 2a_{n-1} + 3a_{n-2} with initial conditions a_0 = 2 and a_1 = 2 to find the closed-form expression.

Let X = {0,1,2,3,4,5,6,7} with addition modulo 8. Verify whether X is a group. Justify your answer.

Define ring and field with their properties and give suitable examples.

Construct a truth table to verify whether the compound proposition not(p implies q) iff (p and not q) is a tautology, contradiction or contingency.

Define a bipartite graph and prove that a simple graph G is bipartite if and only if it contains no odd cycles.

Explain various types of graphs such as simple graph, multigraph, complete graph, bipartite graph and regular graph with suitable examples.

Find the determinant and trace of the matrix A = [[2, -1, 0], [-1, 2, -1], [0, -1, 2]].

Using Cholesky decomposition, verify whether the given symmetric matrix A = [[4, 2, 2], [2, 5, 1], [2, 1, 6]] is positive definite or not.

Find the singular value decomposition (SVD) A = U Sigma V^T of the matrix A = [[1, -1], [-2, 2], [2, 2]].

Explain the concept of an equivalence relation with a suitable example.

What is test of hypothesis? Explain the concept, definition of hypothesis, and null vs alternative hypothesis with formulation.

Write a short note on any two of the following: (i) Time Series Analysis (ii) Analysis of Variance (ANOVA) (iii) Type-II Error.