• Attempt any five questions.
• All questions carry equal marks.
Define Venn diagram. Draw the Venn diagrams for each of these combinations of the sets A, B and C.
i) A ∩ (B - C)
ii) (A ∩ B) ∪ (A ∩ C)
Explain Reflexive, symmetric, Transitive, equivalence relations with example.
Let A = { 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 } be ordered by divisibility. Draw Hasse diagram.
Prove that G = {-1, 1, i, -i} is an abelian group under multiplication.
Find all the subgroups
i) (Z₁₂, +₁₂)
ii) (Z₇*, ×₇)
Solve the recurrence relation aₙ = 2aₙ₋₁ + 3aₙ₋₂ for n ≥ 2 where a₀ = 2 and a₁ = 2.
Define compound proposition. Explain different operations on propositions with truth tables.
Show that (P ∨ Q) ∧ (¬P ∧ (¬P ∧ Q)) ⇔ (¬P ∧ Q).
What is a Graph? Write about various ways of representing graphs.
Using properties of determinants, prove that |a, b, c| |a-b, b-c, c-a| = a³ + b³ + c³ - 3abc |b+c, c+a, a+b|
Find Single Value Decomposition of A = [[1, -1], [-2, 2], [2, 2]]
Explain Type-I and Type-II errors.
Three samples, each of size 5, were drawn from three uncorrelated normal populations with equal variances. Test the hypothesis that the population means are equal at 5% level. Sample 1 | 10 | 12 | 9 | 16 | 13 Sample 2 | 9 | 7 | 12 | 11 | 11 Sample 3 | 14 | 11 | 15 | 14 | 16
Write short note on (any three) a) POSET b) Normal forms c) Gradient of a matrix d) Ring and fields.
