problem 1)a) Let (G, *) be a group. Let H be a non empty subset of G. Show that H is a subgroup if and only if a * b H, (ii) a-1 H, for all a, b H.

b) Demonstrate that every group of order less than or equal to five is abelian.

c) Demonstrate that in any group of even order there is non identity element a which is inverse of itself.

problem 2)a) State Kernel of homomorphism f from a group G,* to a group G ,0. Demonstrate that kernel of f is a normal subgroup of G.

b) Demonstrate that number of partitions of n is equal to number of partitions of 2n with exactly n parts.

c) By generating function technique find an expression for

(i) number of r combinations of n objects with unlimited repetition.

(ii) number of combinations of n with exactly m parts.

problem 3)a) Prove that the number of partitions of n in which no integer occurs more than twice as a part is equal to the number of partitions of n into parts not divisible by 3.

b) Demonstrate that a graph G is a tree if and only if between every two vertices there exists unique path.

problem 4)a) Demonstrate that a connected graph is Eulerian if and only if all of its vertices are of even degree.

b) If the meet operation is distributive over the join operation in a lattice, then show that the join operation is also distributive over the meet operation.

c) State principle of inclusion and exclusion. Determine the number of permutations of n objects 1, 2, …..,n in which no object occupies its proper place. What happens as n tends to infinity?

problem 5)a) Let E x₁,x₂,x₃ x₁ x₂ x₁ x₃ x₂ x₃ be a Boolean algebra . prepare E x₁,x₂,x₃ in both disjunctive and conjunctive normal forms.

b) Prove that

(?x) (P(x) Λ Q(x)) => ( ? x) (P(x) Λ ( ? x) Q(x) . Given suitable ex to show that this converse is no true.