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 x_{1},x_{2},x_{3} x_{1} x_{2} x_{1} x_{3} x_{2} x_{3 }be a Boolean algebra . prepare E x_{1},x_{2},x_{3} 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.