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ANAND INSTITUTE OF HIGHER TECHNOLOGY

DISCRETE MATHEMATICSUNIT – I LOGIC AND PROOFS

PART - A

1. Write down the two propositions P and Q for which Q P is true but P Q is false.

Sol. When Q is false and P is true, we get Q P is true and P Q is false. P: New Delhi is capital of India Q: 4 is an odd number.

2. State the truth value of “If tigers have wings then the earth travels round the sun”.Sol. Let P: Tigers have wings have truth value F Q: The earth travels round the sun have truth value T Therefore, P Q has the truth value T.

3. Construct truth table for (P Q) (P Q).Sol.

P Q P Q P Q (P Q) (P Q)T T T T TT F F F TF T T F FF F T F F

4. Prove that (P Q) (P Q) is a contradiction.Sol.

P Q P Q P Q (P Q) (P Q) (P Q)T T T T F FT F F T F FF T F T F FF F F F T F

(P Q) (P Q) is a contradiction.

5. Give an example of tautology and contradiction.Sol. P (P) is a tautology and P (P) is a contradiction.


6. Prove that (P Q) P Q is a tautology.Sol.

P Q P Q P Q (P Q)(1)

P Q(2)

(1) (2)

T T F F T F F TT F F T T F F TF T T F T F F TF F T T F T T T

Since (1) (2) are all true, (P Q) P Q is a tautology.

7. Prove that P Q P QSol.

P Q P P Q(1)

P Q(2)

(1) (2)

T T F T T TT F F F F TF T T T T TF F T T T T

Since (1) (2) are all true, it is tautology. (1) (2)

(i.e) P Q P Q

8. Prove that P (Q R) ( P Q) R.Sol. P (Q R) P (7Q R) 7P (7Q R) (7P 7Q) R 7(P Q) R (P Q) R.

9. Find the converse and contrapositive of the implication “If it is raining then I get wet.”Sol. Let P : It is raining and Q : I get wet Then the given statement is P Q . The converse of P Q is Q P The contrapositive of P Q is Q P


10. Prove that Q ( P Q) PSol. Assume Q ( P Q) is true. (i.e) Both Q and P Q is true If Q is true then Q is false. and P Q is true, when P is false and Q is false. (i.e) P is false Hence P is true.

11. Define functionally complete set of connectives and give an example.Sol. A set of connectives in which every formula can be expressed in terms of an equivalent formula containing the connectives from this set is called functionally complete set of connectives. Eg : P Q (P Q) (Q P) (P Q) (Q P).

12. Show that is not functionally complete.Sol. Consider the statement P. P cannot be expressed using the connectives , . Hence is not functionally complete.

13. Show that is minimal functionally complete set.Sol. We can express , , in terms of alone. P P (P P) P (P Q) (P Q) (P Q) ((P Q)) P Q

(P P) (P P) P Q (P Q)

((P Q)) P Q

is minimal functionally complete set.

14. Show that is minimal functionally complete set.Sol. We can express , , in terms of alone. P P (P P)


P(P Q) (P Q) (P Q)

((P Q)) P Q

(P P) (P P) P Q (P Q) ((P Q)) P Q

is minimal functionally complete set.

15. Define Principal disjunctive normal form.Sol. For a given formula, an equivalent formula consisting of disjunction of minterms only is known as Principal disjunctive normal form.

16. Define Principal conjunctive normal form.Sol. For a given formula, an equivalent formula consisting of conjunction of maxterms only is known as Principal conjunctive normal form.

17. Define the two rules of inference for statement calculus.Sol. Rule P : A premise can be introduced at any point in the derivation. Rule T : A formula S can be introduced in a derivation if S is tautologically implied by any one or more of the preceding formulas in the derivation.

18. When the set of premises is said to be inconsistent.Sol. The set of premises are said to be inconsistent provided their conjunction implies a contradiction.

19. Show that Q, P Q P.Sol. Argument 1. Q Rule P

2. P Q Rule P 3. Q P Rule T 4. 7P Rule T ( From 1,3)

20. Show that (P Q) follows from P Q by using indirect method.Sol. Argument


1. ((P Q)) Rule P ( assumed premise)2. P Q Rule T3. P Rule T4. P Q Rule P5. P Rule T6. P P Rule Twhich is a contradiction.

21. Symbolize the statement: “If Vani attends classes regularly and if either she is attentive in the class or studies well then she gets the top grade.”Sol. Let P: Vani attends classes regularly.

Q: She is attentive in the class R: She studies well S: She gets the top grade.

The given statement can be written as P (Q R) S.

22. Show that (P Q) (P (P Q)) P Q.Sol. (P Q) (P (P Q)) (P Q) (P (P Q)) (P Q) ((P P) Q)

(P Q) (P Q) [P (P Q)] [Q (P Q)] [(P P) Q] [(Q Q) P] (T Q) (Q P) T (P Q) (P Q).

23. Express the premises in symbols for the following argument: If I like mathematics, then I will study. Either I study or fail. Therefore, if I fail then I do not like mathematics.Sol. Let P: I like mathematics

Q: I will studyR: I failThe given premises are P Q, Q R R P.

24. Find PCNF of P Q if the PDNF of P Q is (P Q) (P Q)Sol. Given P Q (P Q) (P Q)


Now, A (P Q) (P Q)(A) [(P Q) (P Q)] A [P Q] [P Q]

A (P Q) (P Q), which is PCNF.

25. Prove that P Q Q P.Sol. P Q P Q

Q P (Q) P Q P.

26. Define Tautology.Sol. A statement formula which is True regardless of the truth values of the statements which replace the variables in it is called a Tautology. (i.e) The proposition P (P1,P2,……) is a tautology if it contains only T in the last column of its truth values.

27. Define Contradiction.Sol. A statement formula which is False regardless of the truth values of the statements which replace the variables in it is called a Contradiction. (i.e) The proposition P (P1,P2,……) is a contradiction if it contains only F in the last column of its truth values.

28. Test whether the following formula P (Q P) is a tautology?Sol. P (Q P) P (Q P)

P (P Q) (P P) Q

T Q T

Hence P (Q P) is a tautology.

29. Show that (P Q) P QSol. (P Q) [(P Q)]

[P Q]P Q.


30. Establish P Q (P Q) (P Q)Sol.

P Q P Q P Q P Q P Q (P Q) (PQ)

T T T T F F F TT F F F F T F FF T F F T F F FF F T F T T T THence P Q (P Q) (P Q).

31. Define Predicate CalculusSol.: The logic based upon the analysis of predicates of any statement is called Predicate logic (or) Predicate calculus.

32. Define simple statement function with an exampleSol. It is defined to be an expression consisting of a predicate symbol and an individual variable. Ex: Let B be the predicate “is a Bachelor” and j the name “Mr.John”, c “Mr. Chandran” and s “Mr. Senthil”, then B(j), B(c) and B(s) are all simple statement function.

33. Define compound statement function with an exampleSol. It is obtained by combining one or more simple statement function with logical connectives.

Ex: M(x) H(x), M(x) H(x), M(x) H(x), 7H(x), etc.

34. Define Universal Quantifier with an exampleSol. The Quantifiers which are introduced to symbolize expressions such as “for all”, “every” and “for any” is called Universal Quantifier.

Ex: All roses are red. (i.e) If x is rose then x is red. Let A(x): x is rose, B(x): x is red.

Then the statement “All roses are red” can be written as “For all x, if x is a rose then x is red” and it is ( x)[A(x) B(x)] or (x) [A(x) B(x)]

35. Define Existential Quantifier with an exampleSol. The Quantifiers which are introduced to symbolize expressions such as “for some”, “there is at least one” and “there exists some” is called Existential Quantifier. Ex: Some men are clever. Let M(x): x is a man C(x): x is clever


Then the statement “Some men are clever” can be written as “There exists some x such that x is a man and x is clever” and it is ( x) [M(x) C(x)]

36. Write the following statement in the symbolic form “Some monkeys have no tails” Sol. Let M(x): x is a monkey T(x): x has a tail

The given statement can be written as “There is an x such that x is a monkey and x has no tail” and it is ( x) [M(x) 7 T(x)]

37. Write the following statement in the symbolic form“It is not true that all roads lead to Rome”

Sol. Let R(x): x is a road L(x): x lead to Rome

The given statement can be written as “It is not true that for all x if x is a road then x leads to Rome” or “ There is an x such that x is a road and x does not lead to Rome” and it is 7( x) [R(x) L(x)] (or) ( x) [R(x) 7 L(x)].

38. Symbolize : “For any x and for any y , if x is taller than y then y is not taller than x”Sol. Let T(x,y) : x is taller than y. The given statement can be written as “For any x and for any y, if x is taller than y then it is not true that y is taller than x” and it is ( x) ( y)[T(x,y) 7 T(y,x)]

39. Express is an irrational number using quantifiers.Sol. Consider the equation It does not posses any integer solution. Hence we can express is an irrational number as

7( x) ( ).

40. Symbolize: For every x, there exists a y such that Sol. Let P(x,y) : For every x, there exists a y such that can be symbolized as ( x) ( y) P(x,y).

41. Symbolize : For any given positive integer, there is a greater positive integer.Sol. Let P(x): x is a positive integer G(x,y): x is greater than yThe statement can be written as “For any x, if x is a positive integer then there exists some y such that y is a positive integer and y is greater than x” and it is ( x)[P(x) ( y)( P(y) G(y,x))]

42. Symbolize: Some people are not admired by everyone.Sol. Let P(x): x is a person A(x,y): x admires y


The given statement can be written as “There is a person who is not admired by some person” and it is ( x) ( y)[P(x) P(y) 7 A(x,y)]

43. Symbolize: Every book with a blue cover is a mathematics book.Sol. Let P(x) : x is a book B(x) : x has a blue cover M(x) : x is a mathematics bookThe statement can be written as “For all x, if x is a book and x has a blue cover then x is a mathematics book” and it is ( x)[P(x) B(x) M(x)]

44. Symbolize: Every one who likes fun will enjoy each of these plays.Sol. Let L(x) : x likes fun P(y) : y is a play E(x,y) : x will enjoy y.The statement can be written as “For each x, if x likes fun and for each y, if y is a play, then x enjoys y” and it is ( x)( y)[ L(x) P(y) E(x,y)]

45. Symbolize: Every one should help his neighbours or his neighbours will not help him.Sol. Let N(x,y) : x and y are neighbours H(x,y) : x should help y P(x,y) : x will help yThe statement can be written as “For every person x and every person y, if x and y are neighbors, then either x should help y or y will not help x” and it is ( x) ( y)[ N(x,y) ( H(x,y) 7 P(y,x))]

46. Symbolize : Every one who is healthy can do all kinds of work.Sol. Let H(x) : x is a healthy person W(y): y is a kind of work D(x,y): x can do yThe statement can be written as “For all x, if x is healthy and for all y, if y is a kind of work then x can do y” and it is ( x) ( y)[ H(x) W(y) D(x,y)]

47. Symbolize: Some people who trust others are rewarded.Sol. Let P(x): x is a person T(x): x trust others R(x): x is rewardedThe statement can be written as “There is one x such that x is a person, x trust others and x is rewarded” and it is ( x)[P(x) T(x) R(x)]

48. Symbolize: If any one is good then John is good.Sol. Let P(x): x is a person


G(x) : x is good G(j) : John is goodThe statement can be written as “If there is one x such that x is a person and x is good then John is good” and it is ( x)[P(x) G(x)] G(j).

49. Symbolize: He is ambitious or no one is ambitious.Sol. Let P(x): x is a person A(x): x is ambitious ‘He’ represents a particular person. Let that person be y. So the statement is “y is ambitious or for all x, if x is a person then x is not ambitious” and it is A(y) ( x)[ P(x) 7 A(x)].

50. Symbolize: Every student in this class has studied calculus. Sol. Let S(x) : x is a student in this class C(x) : x has studied calculus

The statement can be written as “For all x, if x is a student in this class then x has studied calculus” and it is ( x)[ S(x) C(x)].

PART - B

1. Prove that ((P Q) 7(7P (7Q 7R))) (7P 7Q) (7P 7R) is a Tautology.Sol. ((P Q) 7(7P (7Q 7R))) (7P 7Q) (7P 7R) ((P Q) 7(7P 7(Q R))) 7(P Q) 7(P R) ((P Q) (P (Q R)))) 7[(P Q) (P R)] ((P Q) (P Q) (P R)) 7[P (Q R)] ((P Q) (P R)) 7[P (Q R)] [P (Q R)] 7[P (Q R)] T. Hence the given statement formula is a tautology.


2. Show that 7(P Q) (7P (7P Q)) (7P Q)and hence prove that (P Q) (7P (7P Q)) (7P Q).

Sol. 7(P Q) (7P (7P Q)) (P Q) (7P (7P Q))------(1) (P Q) ((7P 7P) Q)

(P Q) (7P Q) [P (7P Q)] [Q (7P Q)] [(P 7P) Q] [(Q Q) 7P] (T Q) (Q 7P) T (7P Q) (7P Q). From (1) we have (P Q) (7P (7P Q)) (7P Q) Its dual is (P Q) (7P (7P Q)) (7P Q).

3. If H1,H2,…..Hm and P imply Q then H1,H2,…..Hm imply P Q.Sol. We know that A B iff A B is a tautology. Given that H1,H2,…..Hm and P imply Q (i.e) (H1 H2 .….. Hm P) Q

(i.e) (H1 H2 .….. Hm P) Q is a tautology. ----------(1) Since we know that A (B C) (A B) C Equation (1) becomes (H1 H2 .….. Hm ) (P Q) is a tautology (i.e) H1 H2 .….. Hm (P Q).

{Hint: A (B C) A (7B C) 7A (7B C) (7A 7B) C 7(A B) C (A B) C }

4. Obtain PDNF of (P Q) R 7P and hence find its PCNF.Sol. (P Q) R 7P


7[(P Q) R] 7P [7(P Q) 7R] 7P [(7P 7Q) 7R] 7P (7P 7R) (7Q 7R) 7P [(7P 7R) (Q 7Q)] [(7Q 7R) (P 7P)]

[7P (Q 7Q)] (7P 7R Q) (7P 7R 7Q) (7Q 7R P)

(7Q 7R 7P) (7P Q) (7P 7Q) (7P Q 7R) (7P 7Q 7R) (P 7Q 7R) [(7P Q) (R 7R)] [(7P 7Q) (R 7R)] (7P Q 7R) (7P 7Q 7R) (P 7Q 7R)

(7P Q R) (7P Q 7R) (7P 7Q R) (7P 7Q 7R)

(7P Q 7R) (7P 7Q 7R) (P 7Q 7R) (7P Q R) (7P 7Q R), which is PDNF.

Now, 7A (P Q R) (P Q 7R) (P 7Q R) 7(7A) 7[(P Q R) (P Q 7R) (P 7Q R)] A 7(P Q R) 7(P Q 7R) 7(P 7Q R) A (7P 7Q 7R) (7P 7Q R) (7P Q 7R) which is PCNF.

5. Obtain the PCNF of (7P R) (Q P) and hence find its PDNF.Sol. (7P R) (Q P)

(P R) (Q P) (P Q)(P R) (7Q P) (7P Q)[(P R) (Q 7Q)] [(7Q P) (R 7R)]

[(7P Q) (R 7R)](P R Q) (P R 7Q) (7Q P R) (7Q P 7R)

(7P Q R) (7P Q 7R)(P Q R) (P 7Q R) (P 7Q 7R) (7P Q R)

(7P Q 7R) which is PCNF. Now, 7A (7P 7Q R) (7P 7Q 7R) (P Q 7R)

7(7A) 7[(7P 7Q R) (7P 7Q 7R) (P Q 7R)] A 7(7P 7Q R) 7(7P 7Q 7R) 7(P Q 7R) A (P Q 7R) (P Q R) (7P 7Q R) which is PDNF.


6. Using truth tables, verify if 7P is a valid conclusion from the premises 7P Q, 7(Q 7R) and 7R.Sol. Let H1:7P Q , H2: 7(Q 7R) , H3: 7R C: 7P

P Q R C:7P H3:7R H1:7P Q Q 7R H2:7(Q 7R)T T T F F T F TT T F F T T T FT F T F F F F TT F F F T F F TF T T T F T F TF T F T T T T FF F T T F T F TF F F T T T F T

In the 8th row, H1, H2, H3 are all true and C is also true. Hence the argument is valid.

7. Show that P Q, Q 7R, R, P (J S) J SSol. Argument 1. P Q Rule P 2. Q 7R Rule P 3. P 7R Rule T (From 1,2) 4. R 7P Rule T 5. R Rule P 6. 7P Rule T (From 4,5) 7. P (J S) Rule P 8. 7P (J S) Rule T 9. J S Rule T (From 6,8)

The argument is valid.

8. Prove the validity of the following argument: If I get the job and work hard, then I will get promoted. If I get promoted, then I will be happy. I will not be happy. Therefore, either I will not get the job or I will not work hard.


Sol. Let P: I get the job Q: I work hard R: I will get promoted S: I will be happy Then the given premises are (P Q) R, R S , 7S and the conclusion is 7P 7Q. Argument

1. (P Q) R Rule P 2. R S Rule P 3. (P Q) S Rule T (From 1,2) 4. 7S 7(P Q) Rule T 5. 7S Rule P 6. 7(P Q) Rule T (From 5,4) 7. 7P 7Q Rule T


9. Using Rule CP,derive P (Q S) from P (Q R), Q (R S). Sol. Argument

1. P Rule CP (Assumed premise) 2. P (Q R) Rule P 3. Q R Rule T 4. 7Q R Rule T 5. Q (R S) Rule P 6. 7Q (R S) Rule T 7. [7Q R] [7Q (R S)] Rule T 8. 7Q [R (R S)] Rule T 9. 7Q S Rule T 10. Q S Rule T 11. P (Q S) Rule CP The argument is valid.

10. Determine the validity of the following argument: My father praises me only if I can be proud of myself. Either I do well in sports or I can’t be proud of myself. If I study hard, then I can’t do well in sports. Therefore if father praises me, then I do not study well.Sol. Let P: My father praises me


Q: I can be proud of myself R: I do well in sports S: I study hard Given premises are P Q, R 7Q, S 7R and the conclusion is P 7S Argument 1. P Rule CP (Assumed premise) 2. P Q Rule P 3. Q Rule T 4. R 7Q Rule P 5. 7Q R Rule T 6. Q R Rule T 7. R Rule T 8. S 7R Rule P 9. R 7S Rule T 10. 7S Rule T 11. P 7S Rule CP


11. Show that the conclusion R follows from P Q , Q R , P R by using indirect method.Sol. 1. 7R Rule P (Assumed premise) 2. P R Rule P 3. R P Rule T 4. 7R P Rule T 5. P Rule T 6. P Q Rule P 7. Q Rule T 8. Q R Rule P 9. R Rule T 10. R 7R Rule T which is a contradiction.

12. Show that the following set of premises is inconsistent. If the contract is valid then John is liable for penality. If John is


liable for penality then he will go bankrupt. If the bank will loan him money then he will not go bankrupt. As a matter of fact, the contract is valid and the bank will loan him money.Sol. Let P: The contract is valid Q: John is liable for penality R: He will go bankrupt S: The bank will loan him money Given premises are P Q , Q R , S 7R , P S 1. P Q Rule P 2. Q R Rule P 3. P R Rule T 4. S 7R Rule P 5. R 7S Rule T 6. P 7S Rule T 7. 7P 7S Rule T 8. 7(P S) Rule T 9. P S Rule P 10. (P S) 7(P S) Rule T which is a contradiction.

13. Test whether the following formula Q (P 7Q) (7P 7Q) is a tautology or contradiction without constructing the truth table.Sol. Q (P 7Q) (7P 7Q) [(Q P) (Q 7Q)] 7(P Q) [(P Q) T] 7(P Q) (P Q) 7(P Q) T The given formula is tautology.

14. Obtain PCNF of (Q P) (7P Q) and hence find its PDNF.Sol. (Q P) (7P Q) (7Q P) 7P Q (P 7Q) [7P (Q 7Q)] [Q (P 7P)] (P 7Q) (7P Q) (7P 7Q) (Q P) (Q 7P)


(P 7Q) (7P Q) (7P 7Q) (P Q) which is PCNF. Here PDNF does not exists.

15. Obtain PDNF of the formula (P (Q R)) (7P (7Q 7R)). Hence obtain the PDNF of its negation.Sol. (P (Q R)) (7P (7Q 7R)) (7P (Q R)) (P (7Q 7R)) (7P P) (7P 7Q 7R) (Q R P) (Q R 7Q 7R) F (7P 7Q 7R) (P Q R) F (7P 7Q 7R) (P Q R) , which is PDNF. Its negation is (P Q 7R) (P 7Q R) (P 7Q 7R) (7P Q R) (7P Q 7R) (7P 7Q R).

16. Verify the validity of the following argument.“All men are mortal. Socrates is a man. Therefore Socrates is a mortal.”

Sol. Let H(x): x is a man M(x): x is a mortal s: Socrates We need to show ( x)[H(x) M(x)], H(s) M(s) Argument 1. ( x)[H(x) M(x)] Rule P 2. H(s) M(s) Rule US 3. H(s) Rule P 4. M(s) Rule T The argument is valid.

17. Verify the validity of the following argument.Lions are dangerous animals. There are Lions. Therefore, there are dangerous animals.

Sol. Let L(x) : x is a Lion D(x) : x is a dangerous animal We need to show ( x)[L(x) D(x)], ( x) L(x) ( x) D(x) Argument 1. ( x) L(x) Rule P 2. L(a) Rule ES 3. ( x)[L(x) D(x)] Rule P 4. L(a) D(a) Rule US


5. D(a) Rule T 6. ( x) D(x) Rule EG The argument is valid.

18. Verify the validity of the following argument.Every living thing is a plant or an animal. John’s gold fish is alive and it is not a plant. All animals have hearts. Therefore John’s gold fish has a heart.

Sol. Let P(x) : x is a plant A(x) : x is an animal H(x) : x has a heart g : John’s gold fish We need to show

( x)[P(x) A(x)] , 7P(g) ,( x)[A(x) H(x)] H(g) Argument 1. ( x)[P(x) A(x)] Rule P 2. P(g) A(g) Rule US 3. 7P(g) Rule P 4. A(g) Rule T [ From 2,3 i.e.

7P, P Q Q] 5. ( x)[A(x) H(x)] Rule P 6. A(g) H(g) Rule US 7. H(g) Rule T The argument is valid.

19. Verify the validity of the following argument.All integers are rational numbers. Some integers are powers of 2. Therefore, some rational numbers are powers of 2.

Sol. Let P(x) : x is an integer R(x) : x is a rational number S(x) : x is a power of 2 We need to show ( x) [P(x) R(x)], ( x)[P(x) S(x)] ( x)[R(x) S(x)] Argument 1. ( x)[P(x) S(x)] Rule P 2. P(a) S(a) Rule ES 3. P(a) Rule T 4. S(a) Rule T 5. ( x)[P(x) R(x)] Rule P 6. P(a) R(a) Rule US 7. R(a) Rule T ( From 3,6 ) 8. R(a) S(a) Rule T ( From 7,4 ) 9. ( x)[R(x) S(x)] Rule EG.



20. Show that ( x)[P(x) Q(x)], ( x)[R(x) 7 Q(x)] ( x)[R(x) 7 P(x)]

Sol. Argument 1. R(a) Rule CP( assumed premise) 2. ( x)[R(x) 7 Q(x)] Rule P 3. R(a) 7 Q(a) Rule US 4. 7 Q(a) Rule T (From 1,3) 5. ( x)[P(x) Q(x)] Rule P 6. P(a) Q(a) Rule US 7. 7 Q(a) 7 P(a) Rule T 8. 7 P(a) Rule T (From 4,7) 9. R(a) 7 P(a) Rule CP 10. ( x)[R(x) 7 P(x)] Rule UG.

21. Give a proof that the conclusion ( x)[F(x) 7 S(x)] follows from the premises ( x)[F(x) S(x)] ( y)[M(y) W(y)] and ( y)[M(y) 7 W(y)]. Sol. Argument 1. ( y)[M(y) 7 W(y)] Rule P 2. M(a) 7 W(a) Rule ES 3. 7 [M(a) W(a)] Rule T [ P 7Q 7(P Q)] 4. ( y) 7[M(y) W(y)] Rule EG 5. 7( y)[M(y) W(y)] Rule T 6. ( x)[F(x) S(x)] ( y)[M(y) W(y)] Rule P 7. 7( x)[F(x) S(x)] Rule T 8. ( x) 7[F(x) S(x)] Rule T 9. 7 [F(a) S(a)] Rule US 10. F(a) 7 S(a) Rule T[ 7(P Q) P 7Q)] 11. ( x)[F(x) 7 S(x)] Rule UG.

22. Prove that ( x)[P(x) Q(x)] ( x)P(x) ( x)Q(x)Sol. Argument 1. ( x)[P(x) Q(x)] Rule P 2. P(a) Q(a) Rule ES 3. P(a) Rule T 4. Q(a) Rule T 5. ( x)P(x) Rule EG 6. ( x)Q(x) Rule EG 7. ( x)P(x) ( x)Q(x) Rule T.


23. Prove that ( x)[P(x) Q(x)] ( x)P(x) ( x)Q(x)Sol. Argument 1. ( x)[P(x) Q(x)] Rule P 2. P(a) Q(a) Rule US 3. P(a) Rule T 4. Q(a) Rule T 5. ( x)P(x) Rule UG 6. ( x)Q(x) Rule UG 7. ( x)P(x) ( x)Q(x) Rule T.

24. Prove that ( x)[P(x) Q(x)] ( x)P(x) ( x)Q(x)Sol. We shall use indirect method. Argument 1. 7[( x)P(x) ( x)Q(x)] Rule P (assumed premise)

2. 7( x)P(x) 7( x)Q(x) Rule T 3. 7( x)P(x) Rule T 4. 7( x)Q(x) Rule T 5. ( x)7P(x) Rule T 6. ( x)7Q(x) Rule T 7. 7P(a) Rule ES 8. 7Q(a) Rule US 9. 7P(a) 7Q(a) Rule T 10. 7[P(a) Q(a)] Rule T 11. ( x)[P(x) Q(x)] Rule P 12. P(a) Q(a) Rule US 13. [P(a) Q(a)] 7[P(a) Q(a)] Rule T which is a contradiction.

25. Prove that ( x)P(x) ( x)Q(x) ( x)[P(x) Q(x)]Sol. We shall use indirect method. Argument

1. 7( x)[P(x) Q(x)] Rule P (assumed premise) 2. ( x)7[P(x) Q(x)] Rule T 3. ( x)[7P(x) 7Q(x)] Rule T 4. 7P(a) 7Q(a) Rule T 5. 7P(a) Rule T 6. 7Q(a) Rule T 7. ( x)7P(x) Rule EG 8. ( x)7Q(x) Rule EG 9. ( x)7P(x) ( x)7Q(x) Rule T 10. 7( x)P(x) 7( x)Q(x) Rule T 11. 7[( x)P(x) ( x)Q(x)] Rule T


12. ( x)P(x) ( x)Q(x) Rule P 13. [( x)P(x) ( x)Q(x)] 7[( x)P(x) ( x)Q(x)] Rule T which is a contradiction.

26. Prove that ( x)[P(x) Q(x)], ( y)P(y) ( z)Q(z) by using indirect method.Sol. Argument 1. 7( z)Q(z) Rule P (assumed premise) 2. ( z)7Q(z) Rule T 3. 7Q(a) Rule US 4. ( y)P(y) Rule P 5. P(a) Rule ES 6. P(a) 7Q(a) Rule T 7. 7[P(a) Q(a)] Rule T 8. ( x)[P(x) Q(x)] Rule P 9. P(a) Q(a) Rule US 10. [P(a) Q(a)] 7[P(a) Q(a)] Rule T which is a contradiction.

27. Verify the validity of the following inference.If one person is more successful than another, then he has worked harder to deserve success. John has not worked harder than Peter. Therefore John is not more successful than Peter.

Sol. Let S(x,y) : x is more successful than y W(x,y) : x has worked harder than y to deserve success. a : John b : Peter We need to show ( x) ( y)[S(x,y) W(x,y)] , 7W(a,b) 7S(a,b) Argument 1. ( x) ( y)[S(x,y) W(x,y)] Rule P 2. ( y)[S(a,y) W(a,y)] Rule US1

3. S(a,b) W(a,b) Rule US2

4. 7W(a,b) 7S(a,b) Rule T 5. 7W(a,b) Rule P 6. 7S(a,b) Rule T The argument is valid.

28. Symbolize the expression“x is the father of the mother of y”Sol. We can note that in the expression “x is the father of the mother of y”, in between x and y there is a person. Let z be a person as the mother of y.


Let P(z) : z is a person F(x,z) : x is the father of z

M(z,y) : z is the mother of y“x is the father of z and z is the mother of y” is true only for some z and it is ( z)[P(z)F(x ,z) M(z ,y)].

29. Symbolize the expression “All world loves a lover”Sol. All world loves a lover means that Everybody loves a lover. Let P(x) : x is a person S(x ,y) : x loves y Let y be any person who is a lover. P(y) : y is a person L(y) : y is a lover The statement is “For all x, if x is a person then for all y, if y be any person who is a lover then x loves y” and it is ( x)[P(x) ( y){(P(y) L(y)) S(x,y)}].


UNIT 2 COMBINATORICS

PART A

1. Find the number of non-negative integer solutions of the equation

2. Find the recurrence relation for the Fibonacci sequence.

3. If seven colours are used to paint 50 bicycles, then show that at least 8 bicycles will be the

same colour.4. State the Pigeonhole principle.5. Find the no. of arrangements of the letters in SCIENCE. How many of these arrangements have no adjacent E’s ?6. Find n if 7. Solve the recurrence relation y(k) – 8y(k-1) + 16y(k-2) = 0 for k 2, where y(2) = 16 and y(3) = 80.8. What is the solution of the recurrence relation + 2 ?9. How many students must be in a class to guarantee that atleast two students receive the same score on the final exam if the exam is graded on a scale from 0 to 100 points.

PART B

1. Use mathematical induction to prove that is divisible by 8, for all .2. Solve with . 3. Find the generating function of Fibonacci sequence.4. Determine the number of positive integers that are not divisible by 2, 3 or 5 but divisible by 7.5. A bit is either 0 or 1. A byte is a sequence of 8 bits. Find the number of bytes. Among these

how many are (1) Starting with 11 and ending with 00.

(2) Starting with 11 but not ending with 00 or not starting with 11 but ending with 00?

6. Using mathematical induction, prove that 2 + 22 + …..+ 2n = 2n+1 – 2 for all non-negative integers n.

7. A question paper has 3 parts, Part A, Part B and Part C having 12, 4 and 4 questions respectively. A

student has to answer 10 questions from Part A and 5 questions from Part B and Part C put together

selecting at least 2 from each one of these two parts. In how many ways the selection of questions can

be done?

8. Using generating functions, solve the recurrence relation given that

9. Prove by mathematical induction, that for all n 1, n3 + 2n is a multiple of 3.


10. Using the generating function, solve the difference equation

11. How many positive integers n can be formed using the digits 3, 4, 4, 5, 5, 6, 7 if n has to exceed

5000000?

12. Find the number of integers between 1 and 250 both inclusive that are divisible by

any of the integers 2, 3, 5, 7.

13. Using Mathematical induction show that

14. There are 2500 students in a college, of these 1700 have taken a course in C, 1000 have taken a course

Pascal and 550 have taken a course in Networking. Further 750 have taken courses in both C and

Pascal, 400 have taken courses in both C and Networking, and 275 have taken courses in both Pascal

and Networking. If 200 of these students have taken courses in C, Pascal and Networking

(1) How many of these 2500 students have taken a course in any of these three courses C, Pascal and Networking?

(2) How many of these 2500 students have not taken a course in any of these three courses C, Pascal and Networking?

15. Using generating function solve with 16. A box contains six white balls and five red balls. Find the number of ways four balls can be drawn from the box if (1) They can be any colour. (2) Two must be white and two red.

(3) They must all be the same colour.17. If n Pigeonholes are occupied by (kn+1) pigeons, where k is positive integer, prove that at least one Pigeonhole is occupied by (k+1) or more Pigeons. Hence, find the minimum number of m integers to be selected from S = {1, 2,…,9} so that the sum of two of the m integers are even. 18. Solve the recurrence relation

19. Use mathematical induction to show that n , n 2.

20. Use the method of generating function to solve the recurrence relation given that 21. Prove by the principle of mathematical induction, for ‘n’ a positive integer,

22. Find the number of distinct permutations that can be formed from all the letters of each word (1) RADAR (2) UNUSUAL.23. Solve the recurrence relation, S(n) = S(n-1) + 2(n-1), with S(0) = 3, S(1) = 1, by finding its generating function.24. Use mathematical induction to prove the inequality n<2n for all positive integer n. 25. What is the maximum number of students required in a discrete mathematics class to be sure that at least six will receive the same grade if there are five possible grades A, B, C, D and F?26. Suppose that there are 9 faculty members in the mathematics department and 11 in the computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members from the mathematics department and four from the computer science department? 27. Use generating functions to solve the recurrence relation with the


initial condition 28. State and prove generalized Pigeon Hole Principle. 29. Solve the recurrence relation given that = 1 and = 430. Use mathematical induction to show that n3 – n is divisible by 3, for n Z+

31. Show that among any n+1 positive integers not exceeding 2n there must be an integer that divides one of the other integers.

UNIT 3 GRAPHS

PART A

1. Define strongly connected graph.2. Define isomorphism of two graphs.3. Define complete graph and draw a complete bipartite graph of K2,3 and K3,3

4. Define Pseudo graph.5. Obtain the adjacency matrix of the graph G.

6. Give an example of an Eulerian graph which is not Hamiltonian. 7. If a graph contains 21 edges,3 vertices of degree 4 and other each of degree3, how many vertices do the graph

have?8. Define a Complete graph. Draw .9. When is a simple graph G bipartite? Give an example.10. State the necessary and sufficient conditions for the existence of an Eulerian path in a connected graph.11. Define adjacency matrices with an example.12. State Handshaking theorem.13. Define Pendant vertex in a graph.

PART B

1. Draw the complete graph with 5 vertices A, B, C, D, E. Draw all complete sub graph of with 4 vertices.

2. If all vertices of an undirected graph are each of degree k, show that number of edges of the graph is a multiple of k.

3. Draw the graphs for K5 , C5

4. If the simple graph G has 6 vertices and 7 edges then how many edges does have?

5. Determine whether the following graphs are isomorphic. If the graphs are not isomorphic, explain an invariant that the graphs do not share.


6. Prove that in any graph G, the number of vertices of odd degree is even.7. (i) Draw the graph G whose incidence matrix is given below

(ii) Find the adjacency matrix of the given directed multigraph

8. The adjacency matrices of two pairs of graph as given below. Examine the isomorphism of G

and H by finding a permutation matrix.

9. Examine whether the following pair of graphs are isomorphic. If not isomorphic, give the reasons. V1 v2

u1 u2

v5 v3

u5 u3 u4 v4

10. Let G be a simple indirect graph with n vertices. Let u and v be two non adjacent vertices in G such that deg(u) + deg(v) n in G. Show that G is Hamiltonian if and only if G + uv is Hamiltonian.

11. Find all the connected sub graph obtained from the graph given in the following Figure, by

deleting each vertex. List out the simple paths from A to in each sub graph. B

A C


F D E

12. Determine which of the following graphs are bipartite and which are not. If a graph is bipartite, state if it is completely bipartite.2 G1 G 2 G 3

E FE

A B C D A D A

C B C B D E F

13. Using circuits, examine whether the following pairs of graphs G1 ,G2 given below are isomorphic or not: v1 v 4

A

D D B C v2 v 3

G1 G2

14. Prove that the maximum number of edges in a simple disconnected graph G with n vertices

and K components is .

15. Find an Euler path or an Euler circuit, if it exists in each of the three graphs below. If it does not exist, explain why? A

A B A B

F C B C

C D E G1 E D E D

G2 G 3

16. Prove that a connected graph G is Eulerian if and only if all the vertices are of even degree. 17. Show that the graph G is disconnected if and only if its vertex set V can be partitioned into two nonempty subsets V1 and V2 such that there exists no edge in G whose one end vertex is in V 1and the other in V2. 18. How many paths of length four are there from a to d in the simple graph G given below.

a b

c d

19. Show that the complete graph with n vertices Kn has a Hamiltonian circuit whenever n 3. 20. Determine whether the graphs G and H given below are isomorphic.


u u2 v1

uuuuuu v2 v3

v6

u 4 u3 v5 v4

G H 21. Prove that an undirected graph has an even number of vertices of odd degree. 22. Draw a graph that is both Eulerian and Hamiltonian.

UNIT 4 ALGEBRAIC STRUCTURESPART A

1. Define a semigroup.2. If ‘a’ is a generator of a cyclic group G, then show that is also a generator of G.

3. Prove that the identity of a subgroup is the same as that of the group.4. State Lagrange’s theorem in group theory.5. Define homomorphism and isomorphism between two algebraic systems.6. When is a group (G, ) called abelian?7. If a and b are any two elements of a group (G, ), show that G is an Abelian group if and only if

8. Let (M, , be a monoid and aM. If a invertible, then show that its inverse is unique.

9. Define a commutative ring.10. State any two properties of a group.11. Obtain all the distinct left cosets of {(0), (3)} in the group ( and find their union.

12. Show that the set of all elements ‘a’ of a group (G, ) such that ax = xa for every xG is a subgroup of G.

13. Define semi group homomorphism.14. Give an example of a commutative ring without identity.15. Define Ring and give an example of a ring with zero-divisors.16. How many generators are there in a cyclic group of order 10?17. Show that is not a cyclic group.

18. Show that every cyclic group is abelian.19. Define a Commutative Ring.

PART B1. If (G, ) is an abelian group, show that (ab)2 = a2 b2 2. Show that (Z, +, ×) is an integral domain where Z is the se of all integers. 3. State and prove Lagrange’s theorem.4. If (Z, +) and (E, +) where Z is the set all integers and E is the set all even integers, show that the two semi groups (Z, +) and (E,+) are isomorphic. 5. If is a binary operation on the set R of real numbers defined by a b = a + b + 2ab, (1) Find (R , ) is a semigroup (2) Find the identify element if it exists (3) Which elements has inverse and what are they?6. Define the Dihedral group (D4 ,) and give its composition table. Hence find the identify element and inverse of each element. 7. Show that the Kernel of a homomorphism of a group (G, ) into an another group (H,) is a


subgroup of G. 8. Let (S, ) be a semigroup. Then prove that there exists a homomorphism g : S → SS, where (SS,) is a semigroup of functions from S to S under the operation of (left) composition. ]9. Prove that every finite group of order n is isomorphic to a permutation group of order n.

10. Prove that the order of a subgroup of a finite group divides the order the group. 11. Prove the theorem: Let (G, ) be a finite cyclic group generated by an element a G. If G is

of order n, that is, G = n, then an = e, so that G = {a, a2 ,a3,…,an = e}. Further more n is a least positive integer for which an = e.

12. Let f : G → G’ be a homomorphism of groups with Kernel K. Then prove that K is a normal subgroup of G and G/K is isomorphic to the image of f.

13. If is the operation defined on S = Q × Q, the set of ordered pairs of rational numbers and given by (a, b) (x, y) =(ax, ay + b), show that (S, ) is a semi group. Is it commutative? Also find the identity element of S.

14. Prove that the necessary and sufficient condition for a non empty subset H of a group {G, } to be a sub group is a, bH ab-1 H.

15. Prove that the set Z4 = {[0],[1],[2],[3]} is a commutative ring with respect to the binary operation addition modulo and multiplication modulo +4, 4.

16. If f : G G’ is a group homomorphism from {G, } to {G’, } then prove that for any a G, f(a-1) = [f(a)] -1.

17. Show that if G is a finite group of order n, then = e for any a G.

18. Obtain all the elements of the permutation group ( , ) and construct its composition table. Check

whether ( , ) is an abelian group or not. Justify your answer.

19. Let (G, ) be a group and a G. Let f : G G be given by f(x) = a x , for every x G. Prove that

f is an isomorphism of G onto G.20. Prove that intersection of two normal subgroups of a group (G, *) is a normal subgroup of a group (G, *).

UNIT 5 LATTICE THEORY AND BOOLEAN ALGEBRAPART A

1. In a lattice (L, ), prove that a(ab) = a, for all a, b L.2. Define a Boolean Algebra.3. When is a lattice said to be bounded?4. When is a lattice said to be a Boolean Algebra?5. When is a lattice called complete?6. Let A = {a, b, c} and (A) be its power set. Draw a Hasse diagram of ((A), ).7. Check whether the posets {(1, 3, 6, 9),D} and {(1, 5, 25, 125), D} are lattices or not. Justify your

claim.8. Define Sublattice.9. Show that in a lattice if a b c, then a b =b c and (a b) (b c) = (a b) (b c).10. Show that in a distributive lattice, if complement of an element exists then it must be unique. 11. Obtain the partial ordering represented by the Hasse diagram.12. Give an example of a lattice which is modular but not distributive.


13. Draw Hasse diagram of all lattices with upto five elements.14. Which elements of the poset {(2, 4, 5, 10, 12, 20, 25), /} are maximal, and which are minimal?15. Give an example of a distributive lattice but not complemented.

PART B1. Let R be a relation on a set A. Then define R-1 = {(a, b) A A/(b, a) R}. Prove that if (A, R) is

poset then (A, R-1) is also a poset.2. Is the poset (Z+, /) a Lattice?3. Show that every finite partial ordered set has a maximal and minimal element. 4. Show that if L is a distributive lattice than for all a, b, c L,

(ab) (bc) (ca) = (a b)(b c)(c a)5. Show that (N,) is a partially ordered set where N is set of all positive integers and is defined by m

n iff n – m is a non-negative integer.6. In a Boolean Algebra, prove that (ab)’ = a’ b’.7. In a Lattice (L, ), prove that x(yz) (xy)(xz).8. If S42 is the set of all divisors of 42 and D is the relation “divisor of” on S42 , prove that

{S42, D} is a complemented Lattice.9. Prove that Every distributive lattice is modular. Is the converse true? Justify your claim. 10. Show that the direct product of any two distributive lattices is a distributive lattice.11. Draw the Hasse diagram for (1) P1 = {2, 3, 6, 12, 24} (2) P2 = {1, 2, 3, 4, 6, 12} and is a relation

such that xy if and only if x|y. 12. Prove that D110, the set of all positive divisors of a positive integer 110, is a Boolean algebra and find

all its sub algebras. 13. If P(S) is the power set of a set S and , are taken as join and meet, prove that (P(S), ) is a lattice.

Also, prove the modular inequality of a Lattice (L, ), viz for any a, b, c L, a c a (b c) (a b) c.

14. In any Boolean algebra, show that ab’+a’b =0 if and only if a = b.15. Prove that Demorgan’s laws hold good for a complemented distributive lattice (L, , ), viz (ab)’

=a’b’.16. In any Boolean algebra, prove that the following statements are equivalent: (1) a + b = b

(2) a b = a (3) a’ + b = 1 and (4) a b’ = 0 17. Let B be a finite Boolean algebra and let A be the set of all atoms of B. Then prove that the Boolean algebra B is isomorphic to the Boolean algebra P(A), where P(A) is the power set of A

18. Draw the Hasse diagram representing the partial ordering {(A, B):AB} on the power set P(S) where S ={a, b, c}. Find the maximal, minimal, greatest and least elements of the poset.

19. In a Boolean algebra, prove that a.(a+b)=a, for all a, b B. 20. In a distributive lattice {L, , } if an element a L a complement then it is unique.

21. Simplify the Boolean expression a’.b’.c+a.b’.c+a’.b’.c’ using Boolean algebra identities.22. Show that in a lattice both isotone property and distributive inequalities are true.

Also show that in a distributive lattice cancellation law is true.23. Show that in a complemented and distributive lattice a b a = 0 a’ b = 1


b’ a’.24. Let (L, ) be a lattice. If we define a b = glb(a, b) and a b = lub(a, b) for a, b L, then show that both

and satisfy commutative, associative, absorption and idempotent laws.25. Show that Demorgan’s law are true in a complemented and distributive lattice.26. Show that every chain is a lattice.

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