Anna University Coimbatore 2009 B.E Computer Science and Engineering 2 ks with Answer for Discrete Mathematics ( Unit-1) - Question Paper

UNIT I

PROPOSITIONAL CALCULUS

PART A

 

1.     Express the statement Good food is not cheap in symbolic form.

Solution : P : food is Good Q : food is Cheap

Symbolic form is

 

2.     Obtain PDNF for .

Solution : PDNF is

 

3.     If are statement variables , prove that .

Solution : Consider

T

 

4.     Prove that whenever ,we also have and vice versa.

Proof:

To prove we have to prove is a Tautology

is a Tautology

is a Tautology

is a Tautology

is a Tautology

is a Tautology. Hence the proof.

 

5.     Define functionally complete set of connectives and give an example.

A collection of logical operators is called functionally complete if every compound proposition is logically equivalent to a compound proposition involving only those logical operators

 

 

6.     Define Contra positive of a statement.

For any statement formula, the statement formula is called its

converse, is called its inverse and is called its contrapositive.

 

 

7.     Give the converse and the Contra positve of the implication If it is raining then I get wet.

Solution :

P : It is raining Q : I get wet

Converse : : If I get wet, then it is raining

Contrapositive : : If I do not get wet, then it is not raining

8.     Show that . (Use only the laws).

Solution :

(Associative law)

(Distributive law)

(Negation law)

(Domination law)

(Identity law)

 

9.     Write the following statement in symbolic form

(a) Mark is poor but unhappy

(b) mark is rich or unhappy

(c)Mark is neither rich nor happy

(d) Mark is poor (or) he is both rich and poor.

Solution :

M : Mark is poor H : Mark is happy

The symbolic forms are

(a) (b) (c) (d)

10.                        Write the following statement in symbolic form

x is the father of mother of y .

Solution :

P(x) : x is a person

F(x,y) : x is the father of y

M(x,y) : x is the mother of y

The symbolic form is

 

 

 

 

11.                        Write in symbolic form the statement The house will be destroyed if

there is a flood.

Solution :

P : House will be destroyed Q : There is flood

The symbolic form is .

 

 

12.                        Construct the truth table for .

P	Q
T T F F	T F T F	T F T T	T T F T	T F F T

 

 

 

 

 

13.                        Write the dual of .

Solution :

Dual of is

 

 

14.                        Represent using only.

 

15.                        Write the converse, inverse and contra positive of the following

If today is labour day , then tomorrow is Tuesday.

Solution :

P : Today is Labour day Q : Tomorrow is Tuesday

Converse : : If tomorrow is Tuesday then today is labour day.

Inverse : : If today is not labour day then tomorrow is not Tuesday.

Contrapositive : : If tomorrow is not Tuesday then today is not labour day

 

 

16.                        For any statements prove that

.

Solution :

 

 

17.                        Determine the truth value of the following

a) If 3+4=12 , then 3+2=6.

b) If 3+3=6 , then 3+4=9.

a) Here 3+4=12 : F & 3+2=6 : F. ie so truth value is True

b) Here 3+3=6 : T & 3+2=6 : F. ie so truth value is False

 

 

18.                        Write the dual of (a) (b)(c) .

Solution :

a) b) c)

Dual : Dual : Dual :

 

19.                        Show that are not functionally complete set.

Solution :

cannot be expressed using the connectives , since no such contribution of statement exist with as input is T and the output is F.

 

20.                        Express interms of only.

Solution :

 

21.                        Show that .

 

 

 

 

22.                        Demonstrate that is a valid inference from the premises .

Step	Derivation	Rule
(1)		P
(2)		P
(1,2)		T
(3)		P
(1,2,3)		T

 

23.                        Show that .

Step	Derivation	Rule
(1)		P
(2)		P
(1,2)		T
(1,2)		T

 

 

24.                        Show that a)

b) .

 

a ) b)

 

 

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