Question paper 1

1. SECTION-A

   1. Write briefly:
   1. Let

   Answer:

   

   

   

   

2. (b) Let =  find the number of elements of a.

   Answer:

   

   No of elements of A=3.

3. (c) Define an ant is ymmertic relation by giving suitable example .

   Answer:

   Antisymmetric  Relation :A binary relation on a set x is antisymmetric

   If there is no fair of distinct  elts of X each of which is related by R to the other ,

   Or equivalent ly,

   If  

   For example ;

   Is anti cymmetric on the set

   

4. (d)   and let R be a relation from A to B Defined by   Determine the domain and range f R.

   Answer:

   

   

   

   and Range of  

6. (e) Write down the truth table of :

   Answer:

   p	q
   T	T	F	F	F
   T	F	F	T	T
   F	T	T	F	T
   F	F	T	T	T

7. (f) Write down the contrapositive of the conditional preposition; 

   Answer:

   Contrapositive of

     is

    

8. (g) Define a multi graph.

   Answer:

   Multi Graph: A graph which has multiple edges i. is known as multi Graph .

9. (h) Define a simple path and cycle in  a graph.

   Answer:

   Simple path : An open walk in which every vertex is distinct is know as simple path

   Cycle : A closed walk in which no vertex ( except the initial and terminal vertex) appears more than once.

11. (i) Determine whether the sequence  is solution of recurrence relation

    Answer:

      ?

      

    

    

    

      not true

     is not a solution.

12. (j) Find the values of a,b,c,d from the equation: 

    Answer:

    

    

    

    

    

    

    

    

    

    

    

    

13. SECTION-B

    2. IF A and B are any two sets, then prove that 

    Answer:

    

    

    

    

    

    

14. 3. Prove the following by the principle of  mathematical induction

           

    Answer:

     

    Basic we prove the result for n=1.

    

    

     

     Assumption : We assume the result is true for n=k.

    

    

    

    

    

    

    

      

     

16. 4. (a) Define the following graphs by taking suitable examples .

    1. Eulerian graph.

    Answer:

     A graph which contains either Euler path or Euler

    Circuit is called Eulerian Graph.

     

17. ii. Hamiltonian graph

     

    Answer:

     A Graph Which contains either Hamiltonian circuit a Hamiltonian graph

18. (b) find the minimum n of colors required to paint the following graph

    

    Answer:

19. 5. Find the inverse of the following matric

                       

    Answer:

    let  

    

    

    

           Co- factor matriy= 

    

    

21. 6.  (a) Consider the following three  relations on the set

    

    

     

    Determine whether or not each of the above relations on A is:

    (i) Reflexive      (ii)  symmetric;  (iii) transitive;

    Answer:

    (a)

    i. the only T is reflexive

    

     

    ii.  

    .

    

           S is not sym .

    (iii)

22. (b) Verify that the proposition  is a contradiction.

    Answer:

    

    P	Q	p∧q	p∨q	∼(p∨q)	(p∧q)∧∼(p∨q)
    T	T	T	T	F	F
    T	F	F	T	F	F
    F	T	F	T	F	F
    F	F	F	F	T	F

     

    

23. 7. (a) Determine which of the following are Eulerian or Hamilton or both?

    

    Answer:

    

     

    This graph contains vertices of add degree.

    ∴ Graph is not evler graph Hamiltonian circuit 

24. (b) In a group of 50 person 14 drink tea but not coffee and 30 drink tea fond:

    (i) How many drink tea and coffee both? (ii) How manu drink coffee but not tea ? 

    Answer:

    

     

     

    The graph is Euler Graph

    In this graph, neither Hamiltonian circuit exists nor the Hamiltonian path

     Graph is not Hamiltonian Graph.