BC0039Discrete Mathematics[Part A 1 mark each]1) If a set A has n elements then its power set P(A) has _______ elements. a) 2n b) 2n c) 2+n d) 2-n 2) Difference of two set A & B i.e. A-B is the set of elements of ________. a) b) c) d) A which are in B B which are not in A A which are not in B None of the above

3) For any three sets A, B, C , A (BUC) = a) b) c) d) (A-B) U (A-C) A (BC) (A - B) UC (A-B) (A-C)

4) R = { (2, 4), (2, 6), (3, 6), (3, 9) } domain of R isa) b) c) d) {2, 3} {4, 6, 9} {2, 3, 4, 6, 9} None of the above

5) If a relation R in a set A is distributive if (a, b) R, (b,c) R it implies a) b) c) d) (a, b) R (b, c) R (a, c) R (c, a) R

6) R is equivalence relation ifa) b) c) d) R is reflexive R is symmetric R is transitive All of the above

7) Which of them following is an open interval. a) b) c) d) [a, b) [a, b] (a, b) None of the above

8) A G.P has the form. a) b) c) d) a, a+d, a+2d _ _ _ _ _ _ a, ar, ar2 , ar2 1/a, 1/a+a, 1/a+2d _ _ _ _ _ a, (a+d) r, (a+d) r2 _ _ _ _ _ _

9) Value of p (n, r) is a) b) c) d) n!/r! (n-r) ! r!/(n-r) ! n!/(n-r)! (n-r) !/n!

10) Value of 99.1 is a) 99 b) 98 c) 100 d) 99.1 11) Value of 3.9 is a) 3 b) 4 c) 2 d) 3.9 For Que 12-15 S = {1, 2, 3, 4, 5, 6, 7, 8}, S1 = {1, 2}, S2 = {3, 4, 5} 8 6 4 3 1 2 7 5

12) Glb of S1 a) {1, 2} b) {3, 4, 5} c) {3, 4, 5, 6, 7, 8} d) None 13) Lub of S2 a) b) c) d) {1, 2} {6, 7, 8} {3} None

14) Lower bound of S1. a) b) c) d) {1, 2} {1, 2, 3, 4} {4, 5, 6} None

15) Glb of S2. a) {1, 2} b) 3 c) {4, 5, 6, 7, 8} d) None 16) Lub of S1 a) {1, 2} b) {1, 2, 3, 4} c) {4, 5, 6} d) None 17) Upper bound of S1 a) 3,4,5,6,7,8 b) 1,2,3 c) 6,7,8, d) None 18) Upper bound of S2 a) 3,4,5,6,7,8 b) 1,2,3 c) 6,7,8 d) None 19) (A, *) be an algebraic system where is as binary operation on A (A, *) is called a semigroup if-

a) b) c) d)

* is a closed operation * is an associative operations None of the above All of the above

20) (A, *) be an algebraic system where * is a binary operation on A. (A, *) is called monoid if it satisfy. a) b) c) d) * is a closed operation * is an associative operation Existence of identify All of the above

21) A non empty set G together with a binary operation * is called a group if the algebraic system (G, *) satisfies. a) b) c) d) e) (a, b) are elements of G, implies (a*b) is an element of G (a*b)*c = a* (b*c) for all a, b, c G There exists are element e in G such that a*e =e*a= a for all a G For any element a m G there correspond an element b in G such that a * b = b *a All of the above

22) Which of the following is not a proportion. a) b) c) d) Bangalore is the capital city of Karnataka. 3+6=7 x+y+5=9 Berlin is the capital of Australia.

23) 53. [p (pq)] p isa) b) c) d) Tautology Proposition Contradiction Statement

Consider for Que 21-22 p= This laptop is good, q= This laptop is cheap. The symbolic form for the statement. 24) The laptop is not good but cheap. a) p q b) (~p) q

c) (~ p) (~q) d) (pq) 25) This laptop is neither good nor cheap. a) b) c) d) (p q) pq ~ p ~q pq

26) The predicate in the statement. Stefan is a Mathematician a) b) c) d) Stephen is a Stephen is a Is a mathematician

27) Bhanu is taller then Prajwal. a) b) c) d) Bhanu Prajwal is taller than taller than Prajwal

28) p (x) = x>3. P(2) = ? Truth value. a) b) c) d) F T Either F & T Neither F nor T

29) Write ~p if p is - x>20 a) b) c) d) x=20 x1 , ao =1 a) 3n b) 3n c) 3+n d) 3-n 48)[p (pq)] p isa) Tautology b) Proposition c) Contradiction d) Statement e) 49)Consider the following statement Some men are clever is written asa) b) c) d) ( x) (m (x)) (x (m (x) (C (x)) (x) (m(x)) ((x) ( x) (m(x) ((x)

50)Some real no. are rational. a) ( x) ((R (x) (x)) b) (x) (R(x) c) (x) (R(x) (x)) d) (x) (R(x) (x)) 51)Convert the following statements into symbolic logic. All beautiful birds are ornately colored a) b) c) d) (x) ((D(x) B(x) O(x)) (x) ((D(x) B(x) O(x)) (x) ((D(x) B(x) O(x)) (x) ((D(x) B(x) O(x))

52) Birds do not live on honey are dull in color. a) b) c) d) (x) ((D(x) H(x) O(x)) (x) ((D(x) H(x) O(x)) (x) ((D(x) H(x) O(x)) (x) ((D(x) H(x) O(x))

53)A= {1, 2, 3, 4, 5, 6} the relation / divides is a partial order relation on A The hasse diagram of (A, /) is

a. 5

6 3

b. 4 2 5 1 3

6

1

2

4 c. 6 2 6 3 d. 4

5 3

2 1 5 1 54)Determine which of the following are partial order? a. d. g. h. R1 = {(a, b) z * z / b. a-b c. 1 } on z R2 = {(a, b) z * z / (a) e. b f. } on z R3 = {(a, b) z * z / a divides b in z} on z R4 = {(a, b) z *z / a-b 0}

55)(A, *) be an algebraic system where * is a binary operation on A (A, *) is called a semi group if it satisfy. a. * is a closed operation i.e. a*b A for all a, b A b. * is an associative operation i.e. a* (b*c) = (a*b) * c for all a, b, c A c. All of the above d. None of the above 56)* on A = {a, b, c, d} * a b c d

a b c d

d a a a

a b b b

b c c c

c d c d

Left identify isa) b) c) d) a&b a&c b&d b&d

57) O on A = {a, b, c, d} O a b c d a a b c d b b a d a c c c a b d d d b c

Right identify is > a) b) c) d) a b c d

Product of Transposition and also show whether which is true for even or odd 58) 123 213 a. b. c. d. 59) 1234 4321 a. (1, 2) (3, 4) odd b. (1, 2) (4, 3) even c. (1, 4) (2, 3) odd Odd (1, 2) Even (1, 2) Odd (2, 3) Even (2, 3)

d. (1, 4) (2, 3) even 60) If f(x) = x +3 and g(x) = 4x -5; then g of is2

a) b) c) d)

1 4x2 + 3 -5 none

[Part C 4 marks each]61)F(x) = 1/1-X, X 1 , F [F{F(X)}] = ? X2 b) X/X-1 c) X-1/X d) Xa)

62) If Tp = q, Tq = p a) b) c) d)63)

than Tp+q = ?

0 p+q p q

Using mathematical induction solution of the series. 13 + 23 + 33 + _ _ _ _ _ _ + x3 a) b) c) d) n2 (n+1) 2/2 n3 (n2+n+1)/3 n2 (n+1) 2/4 n(n+1)/2

64)Sum of n bracket series is (1) + (1+2) + (1+2+3) _ _ _ _ _ _ a) b) c) d) n(n+1) (x+2) (x+3)/127 n(n+1) (x+2)/4 n(n+1)/2 n(n+1) (n+2)/6

65)Sum of to infinity the series. 1 + 4/9 + 7/92 + 10/93 + _ _ _ _ _ _ _ a) b) c) d) 99/66 77/64 99/81 99/64

66)How many solution does the eg. x+y+z = 77 have where x, y, z are non negative integers? a) b) c) d) 170 171 100 271

67)Solution of the recurrence relation An = an-1+2an 2 with a0 =2, a1=7 a) b) c) d) 2-2n-(-1)n 3.2n-(-1)n 2n (-1)n 4.2n (-1)n

68)P(x) : x is a person F(x, y)= x is the father of y M(x, y)= x is the mother of y Write the predicate X is the father of the mother of y a) b) c) d) z (p(z) f(x, y) (m(x, y)) z (p(z) f(x, z) (m(z, y)) z (p(z) f(x, z) (m(x, z)) z (p(z) f(x, z) (m(z, y))

69)(p (p q) p) is a : d) e) f) g) Tautology Contradiction Statements None of the above

70)How many slices. of pizza can a person obtain by making n straight into with a pizza knife a) x(x=1) 2 b) n(n+1) 2 c) n(n+1) (n+2) 6 d) n(n+1) 4 For Que 11-12

f(x) =1+x+x+x+x4+x5----------+xn+----------g(x)= 1-x+x-x+x4-x5----------+(-1)nxn+----------71)Find f(x) + g(x) is a) b) c) d) 2+2x+2x+2x+-----------2x+4x+6x+ -----------2+2x+2x4+----------1+x+x2+x3+x4---------

72)f(x) . g(x). For the function under is a) b) c) d) 1+x2+x4+x6+-------1+x+x2+x3+x4------x+x2+x3+x4-----------2x+2x2+2x3+2x4+-----------

== 73)h)

1234 4123 1234 2143 then

=b. 1234 4213 d. 1234 3124

1234 3214 1234 1432

c)

74)

a) 1234 3214 1234 b. 1234 4213 d. 1234

c)

1432

3124

75) Match the following If G is a Group and H is subgroup, aG and hH 1. Cyclic Group 2. Cyclic Subgroup 3. Left Coset 4. Right Coset a) b) c) d) (1,1),(2,2),(3,3),(4,4) (1,1),(2,4),(3,3),(4,2) (1,3),(2,4) (3,1),(4,2) None of these 1. then (a)={ ai/i= 01,.} generated by the element a 2.Ha is right coset 3. aH is Left coset 4. G=(a)