Foundation of Computer Science

(b) Show that the intersection of 2 normal subgroups of a group G is also normal subgroup G.

Prove the following:-

(c) 

Prove the following:-

(b) 

Prove the following:

(a) 

(a) Define the connectives conjunction and disjunction and give the truth table for pVq.

(e) Define function, domain, Co-domain and range of a function. 

(d) Find DNF for the function F(x, y, z) = (x + y)z'.

(c) State and prove the De Morgan's law for a Boolean algebra.

(b) Determine the contrapositive of the statement "If John is a poet, then he is poor."

(c) If all the vertices of an undirected graph are each odd degree k, show that the number of edges of the graph is multiple of k. 

(b) Give an example of graph which contains a Hamiltonian circuit, butnot a Eulerian Circuit. 

(a) Define Hamiltonian Circuit with Example.

(a) Explain the principle of mathematical induction.

(a) Explain Lagrange's Theorem with proof.

(c) Explain principle of inclusion and exclusion with an example.

(b) Define lattice and give an example. 

(a) Draw the Hasse diagram for the divisibility for the divisibility relationon [2,4,5,10; 12, 20, 25} starting from the digraph. 

(c) Prove that the group (G,+6) is a cyclic group where G = {0,1,2,3,4,5).

(b) Define cyclic group with example,

(a) Show that all proper subgroups of groups of order 8 must be abelian.

(b) Show that the minimum number of edges in a connected graph with n vertices is (n-1).

(a) Explain Vertex coloring problem and chromatic number of graph using example. 

(c) What is extended pigeonhole principle, explain with suitable example.

(b) Explain the Partial Ordered Relation with the help of suitable example,