Set - I School of Science and Humanities

Subject : Higher Mathematics (CSE 501)

Course Teacher: N.Chandramowliswaran

M.Tech Information Technology[Networking]-2009 Batch: IT 2

Time: 1 Hour Max.Marks: 20

Answer all the following Questions 20 x 1 = 20 Marks

1. Say True or False ( ) ( ) ( )P Q P Q P Q P Q ¬ ∨ ⇔ ¬ ∧ ∨ ¬ ∧¬ ∨ ∧

2. ( ) ( ) ( ) ( ) ( ) ( )P Q P R Q R P Q R P Q R P Q R ___?___ ∧ ∨ ¬ ∧ ∨ ∧ ⇔ ∧ ∧ ∨ ∧ ∧¬ ∨ ¬ ∧ ∧ ∨

a. ( )P Q R¬ ∧¬ ∧ b. ( )P Q R¬ ∧ ∧¬ c. ( )P Q R∧¬ ∧¬ d. ( )P Q R¬ ∧¬ ∧¬ 3. ( ) ( ) ( )P P Q P Q ___?___ P Q ∨ ¬ ∧ ⇔ ∧ ∨ ∨ ¬ ∧

a. ( )P Q¬ ∧ b. ( )P Q¬ ∨ c. ( )P Q∧¬ d. ( )P Q∨¬ 4. Among the three statement formulae which of them forms a tautology.

(i) ( ) ( )Q P Q P Q∨ ∧¬ ∨ ¬ ∧¬

(ii) ( )Q P Q∧ ∨¬

(iii) ( )( )P P Q P→ ∧ → a. (i) & (ii) b. (i) & (iii) c. (ii) & (iii) d. (i), (ii) , (iii) 5. Among the three statement formulae which of them forms a tautology.

I. P Q P Q ∧ ⇒ ∧ II. ( ) ( )P P Q Q ¬ ∧ ∨ ⇒

III. ( )P P Q Q ∧ → ⇒ a. I & II b. I & III c. II & III d. I , II , III e. None of the above

6. Out of two statement formulae which of them forms a tautology.

I. ( ) ( ) ( )P Q R P R Q→ ⎯⎯→ ∨ → ∨⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

II. ( ) ( ) ( )P Q R P R Q→ ⎯⎯→ ∧ → ∧⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ a. I b. II c. I & II d. None of the above

7. Among the three statement formulae which of them forms a tautology.

I. ( ) ( )Q P Q P ¬ ∧ → ⇒ ¬

II. ( ) ( ) ( )P Q Q R P R → ∧ → ⇒ →

III. ( ) ( ) ( )P Q P R Q R R ∨ ∧ → ∧ → ⇒ a. All the above b. None of the above c. I & II d. II & III

8. Out of two statement formulae which of them forms a tautology. I. ( ) ( ) ( )P Q R S P S Q S R→ → ⎯⎯→ ∨ → ∨ → ∨⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

II. ( ) ( ) ( )P Q R S P S Q S R→ → ⎯⎯→ ∧ → ∧ → ∧⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ a. I b. II c. I & II d. None of the above

9. Among the three statement formulae which of them forms a tautology.

I. ( ) ( ) ( ) ( )P Q R P S Q S R S S∨ ∨ ∧ → ∧ → ∧ → ⎯⎯→⎡ ⎤⎣ ⎦

II. ( ) ( ) ( ) ( )P Q Q R R S P S→ ∧ → ∧ → ⎯⎯→ →⎡ ⎤⎣ ⎦

III. ( ) ( )Q R R Q¬ → ⎯⎯→ ¬ → a. All the above b. None of the above c. I & II d. II & III

10. Among the three statement formulae which of them forms a tautology.

I. ( ) ( )Q R S U U S R Q⎡ ⎤ ⎡ ⎤¬ → → → ⎯⎯→ ¬ → → →⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

II. ( ) ( ) ( )P Q P P Q P Q⎡ ⎤¬ ∧ → → → ⎯⎯→ →⎡ ⎤⎣ ⎦⎣ ⎦

III. ( ) ( ) ( )P Q P P Q P Q⎡ ⎤→¬ → → → ⎯⎯→ →⎡ ⎤⎣ ⎦⎣ ⎦ a. All the above b. None of the above c. I & III d. II & III

11. 1. ( ) ( )e f b e = ?∧ ∨ ∧ 2. ( ) ( )( )a b a c ?∨ ∧ ∨ =

12. ( ) ( )a c e g f d c ?∨ ∨ ∨ ∧ ∧ ∧ =

I. does not exist II. d III. e IV. c

0

a b

c

b

e f

I

a

0 h

e

c

a b

d

f g

13. Fig 1 Fig 2 Fig 3 Fig 4 Choose the correct statement

I. Figure 1 has both greatest and least element II. Figure 2 has greatest element III. Figure 3 has neither greatest nor least element IV. Figure 4 has greatest element

14. Fig 1 Fig 2 Fig 3

Fig 4 Fig 5 Fig 6 Fig 7

Choose the correct answer. I. Fig 1 and Fig 5 does not form a Lattice, rest all form a Lattice II. Fig 2 and Fig 3 forms a distributive Lattice III. Fig 6 and Fig 7 forms a Lattice but not distributive Lattice IV. Fig 4 forms a distributive Lattice

a b

c

d

e

f

b

c

d

e

f

a

a

cd e

f

b

g h

f

e

c

a

b

d

5

10

20

4

2

1

a

b

c

d

e f h

e

c

a b

d

f g

a

b

c de

a

b c d

e

d

a

b c

e

f g

a

b

c d

e

15. Let ( )L,≤ be a Lattice. Then for every a and b in L,

a. ∨a b = b if and onlf if ________ b. ∧a b = a if and onlf if ________ c. ∧a b = a if and onlf if ________

16. Let L be a Lattice. Then Which of the following statements are true.

I. ∨ ∧a a = a , a a = a II. ∨ ∨ ∧ ∧a b = b a , a b = b a III. ( ) ( ) ( ) ( )∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧a b c = a b c , a b c = a b c IV. ( ) ( )∨ ∧ ∧ ∨a a b = a , a a b = a

17. Choose the correct statement

Let L be a Lattice. Then for every a, b, and c in L , I. ≤ ≤ ∨ ≤a c and b c if and onlf if a b c II. ≤ ≤ ≤ ∧c a and c b if and onlf if c a b III. ≤ ≤ ∨ ≤ ∨ ∧ ≤ ∧If a b and c d , then a c b d , a c b d

18. Let be the set of all integers. Let the standard less than or equal to symbol , " "≤

be the given partial order. Then I. x y _______∧ = II. x y _______∨ =

19. _______ follows logically from the premises ( ) ( )C D,(C D) H, H A B R S∨ ∨ →¬ ¬ → ∧¬ → ∨ a. R S∨¬ b. R S¬ ∨ c. R S¬ ∨¬ d. R S∨ 20. _______ is tautologically implied by ( ) ( ) ( )P Q P R Q S∨ ∧ → ∧ →

a. S R∨ b. S R¬ ∨¬ c. S R¬ ∨ d. S R∨¬