Discrete Mathematics Structure

How many one-to-one functions are there from a set with five elements to a set with 10 elements?

What is a bipartite graph? Give example.

Simplify the expression "(P or Q) and (not P)" using logical operation.

Define homomorphism and isomorphism of groups.

What is a generating function? Give example.

What is the difference between path and circuit? Define hamiltonian path and circuit.

Define the predicates and quantifiers in propositional logic.

Define the properties of equivalence relation.

What is Pigeonhole Principle?

Define the symmetric difference of two sets using venn diagram.

What is the minimum number of students required in a discrete mathematics class to be sure that atleast six will receive the same grade, if there are five possible grades, A, B, C, D and F.

Suppose that number of bacteria in a colony triples every hour : (i) Set up a recurrence relation for the number of bacteria after "n" hours have elapsed. (ii) If 100 bacteria are used to begin a new colony, how many bacteria will be in the colony in 10 hours.

Give the recursive definition a function "L" from the set "s" of strings over {a, b} to the set of natural numbers that gives the length of a string.

Prove by contradiction that √2 is an irrational number.

Given sets A = {1, 2, 3} and B = {2, 3, 4}, find union, intersection and complements of set A and B.

Determine if the set {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20} forms a lattice under the relation "divides".

How many ways are there to arrange the letters in the word "COMBINATIONS" such that all vowels are together?

Find the generating function for the sequence {1, 1, 2, 3, 5, 8,.....} defined by fibonacci numbers.

Answer these questions for the partial order represented by following Hasse diagram.

Show that the function f(x) = |x| from the set of real numbers to the set of non-negative real numbers is not invertible but if the domain is restricted to the set of non-negative real numbers, the resulting function is invertible.

Use Kruskal's algo to find a minimum spanning tree in the weighted graph as shown below.

(i) If G is a Group such that (a, b)^2 = a^2 b^2 for all a, b ∈ G then show that G must be abelian. (ii) Determine whether following posets are lattices.