AI/AL/CD-401 (GS) – Introduction to Discrete Structure and Linear Algebra

If A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}, a ∈ A, b ∈ B, find the set of ordered pairs such that 'a' is factor of 'b' and a < b.

Let R be the relation on the set R of all real numbers defined by a R b if and only if |a - b| ≤ 1. Then prove that R is reflexive, symmetric, but not transitive.

Let A = {1, 2, 3, 4, 5, 6} and a R b if and only if a is multiple of b. Find:

i) Domain

ii) Range

iii) Matrix of a relation

iv) Digraph of the relation R.

What common relations on Z are the transitive closures of the following relations?

i) a S b if and only if a + 1 = b,

ii) a R b if and only if a - b = 2.

i) Prove that p ∧ q ⇒ q ∨ p is a Tautology.

ii) Show that (p ∨ q) ∧ (¬p) ∧ (¬q) is a contradiction.

Prove that in any graph G, even number of vertices is of odd degree.

Solve the equations 6x + 15y + 55z = 76, 15x + 55y + 225z = 295, 55x + 225y + 979z = 1259 using Cholesky decomposition method.

Find singular value decomposition for A = [[1, 0, 1], [-1, 1, 0]].

To test the hypothesis that eating fish makes one smarter, a random sample of 12 persons take a fish oil supplement for one year and then are given an IQ test. Test using the given hypothesis, report the test statistic with the P-value, then summarize your conclusion. H₀: μ = 100, Hₐ: μ > 100.

What are the critical values for a one-independent sample non directional (two-tailed) z test at a .05 level of significance?

Let A = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24} be ordered by divisibility. Draw Hasse diagram.

Let G be a planar graph with 10 vertices, 3 components and 9 edges. Find the number of regions in G.

Find chromatic number of the given graph.

Write short note on partially ordered sets and explain with suitable example.

Write short Note on: a) Hasse diagram b) Lattice c) Weighted Graph