Assignment Questions, 20101 - Nalanda Open University

PAPER-VAnswer Any Two Questions.(lHkh iz'u 10-10 vadska ds gS)1. (a) Prove that lattices of normal subgroups of an arbitrary group is a modular lattice.(b) Give an example of a partially ordered set which is not a lattice.2. (a) Find a necessary and sufficient condition for a lattice to be modular.(b) Prove that a complemented modular lattice is relativelycomplemented.3. (a) Prove that complement of an element in a Boolean algebra is unique.(b) Prove that if B is Boolean algebra and x, y, z ∈ B, thenx ∧ (y - z) = (x ∧ y) - (x ∧ z).PAPER-VIAnswer Any Two Questions.(lHkh iz'u 10-10 vadska ds gS)1. (a) Discuss Dirichlet's problem for a half-plane.(b) State and derive Cauchy integral formula for the half-plane.2. (a) Show that Green's function for I(c) is symmetric.(b) If a I(c), then G(z, a) > o for each z in I(c) such that z a.3. (a) Define an analytic function and obtain Cauchy Riemann Differential equations.(b) Show that the function u = is harmonic and find the corresponding analyticfunction.PAPER-VIIAnswer Any Two Questions.(lHkh iz'u 10-10 vadska ds gS)1. (a) State and prove Ascoli's theorem.(b)Compute the first three successive approximations for the solution of the followingequation .2. State and prove Cauchy-Peano existence theorem.3. (a) Define a linear system and show that it satisfies Lipschitz condition and set of itssolutions form a vector space.(b) Solve the system of linear equationsPAPER-VIIIAnswer Any Two Questions.(lHkh iz'u 10-10 vadska ds gS)1. (a) Show that the set of all real numbers is uncountable.(b) Prove that every set can be well ordered.2. (a) For any cardinal numbers α, β and γ show that(i) α ( β + γ) = αβ + αγ (ii) ( ( (b) State and prove Cantor's theorem.3. (a) Prove that an undirected graph has even number of vertices of odd degree.(b) Let G be a non-directed graph with 12 edges. If G has 6 vertices each of degree 3and the remaining having degree ≤ 3. Find the minimum number of vertices of G.M.C.A Part-IPaper-I(Problem Solving and Programming)Answer Any Two Questions.(lHkh iz'u 10-10 vadska ds gSa1. Explain the different types of data types in 'C', with examples.2. Describe different types of control statements in 'C', with suitable examples.3. What happens if an array is used without initializing it. Describe different types of arrayswith examples.