Define: Boolean algebra. Draw Hasse diagram of (S , D) and determine all the sub Boolean of the Boolean algebra ‹ S , Λ, V, ‘ , 0 , 1 ›.30
[7 marks]Use Predicate, Quantifier and logical connectives, Rules of inferences, Argument forms to prove the following consequence is valid. All athletes are healthy. All healthy people take vitamins. Grant is an athlete. Therefore Grant takes vitamin.
[7 marks]Define Lower bound and greatest lower bound. Let P = < {3, 5, 9, 15, 24, 45},D> be a poset. Draw the Hasse diagram. Find
[7 marks]The maximal element & minimal element. ii) The greatest and the least element. iii)The lower bounds of {3, 5}, if any & the upper bound of { 9, 15 } , if any. iv) GLB of {15, 45} & LUB of {3, 9, 15}.
[ marks]Using proof by contradiction method, prove that 2 is an irrational number.
[7 marks]Define an equivalence relation. Prove that the relation “congruence modulo 3” given by ≡ {<x, y> / x-y is divisible by 3} over the positive integer is an equivalence relation.
[7 marks]Define: Sub lattice. Verify whether the given sets are sub lattice of (S ,D) or not. (i) [{1,2,3,6} ,GCD , LCM] (ii) [{1,3,6, 15 } ,GCD , LCM]
[7 marks]Use the Karnaugh map representation to find a minimal sum-of-products expression of f(a ,b , c , d) = ∑ (0,1, 2,3,13,15).
[7 marks]Define complemented lattice. Which of two lattices <Sn, D> for n=30 and n=45 are complemented? Draw Hasse Diagram of these lattices.
[7 marks]Use the Quine McClusky method to simplify the SOP expansion, f(a ,b , c , d) = ∑ (0, 2,4,6,8, 10, 12, 14).
[7 marks]Define: Group and Abelian group. Show that in a group (G, *), if for every a, b € G, (a * b)2 = a2 * b2, then (G, *) is an Abelian group.
[7 marks]Define :Left coset of a subgroup <H, *> in the group<G, *>. Find left cosets of {[0], [3]} in the group <Z6, +6>. OR1
[7 marks]Define Cyclic group. Show that (Z ٭ , ٭ ) is a cyclic group and find all the generators of (Z ٭ , ٭ ). Where Z٭ = Z -[0]
[7 marks]Show that the symmetric set (S , o) is a group. Is it abelian? Justify.3
[7 marks]Define Nodebase. Find Nodebase of the following digraph.
[7 marks]Define adjacency matrix. Draw di-graph of given adjacency matrix.Find in-degree & out-degree of each vertex from the given adjacency matrix. 1 1 1 A 1 1 1 1 1 1
[7 marks]Define: Strong component, Unilateral component and weak component of the digraph. Determine Strong component, Unilateral component and weak component of the following digraph.
[7 marks]Define tree. Give three different representation of the given tree. (v0(v1(v2)(v3)(v4))(v5(v6)(v7)(v8)(v9))(v10(v11)(v12)))
[7 marks]