QUESTION BANK. 4. State and prove De Morgan s law of set theory. (6m) May / jun 10

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1 UNIT 1: Set Theory QUESTION BANK 1. In a survey of 120 passengers, an airline found that 48 enjoyed wine with their meals, 78 enjoyed mixed drinks, 66 enjoyed iced tea. In addition, 36 enjoyed any given pair of these beverages and 24 enjoyed them all. If two passengers are selected at random from these survey sample of 120, what is the probability that they both want only iced tea with their meals? (7m) may /jun Find the probability of getting a sum different from 10 or 12 after rolling two dice. (5m) may/jun Explain set operations: (8m) may /jun State and prove De Morgan s law of set theory. (6m) May / jun In a survey of 260 college students, the following data were obtained: 64 had taken a mathematics course, 94 had taken a computer science course, 58 had taken a business course, 28 had taken both a mathematics and a business course, 26 had taken both a mathematics and a computer science course, 22 had taken both a computer science and a business course, and 14 had taken all three types of courses. a. How many of these students had taken none of the three courses? b. How many had taken only a computer science courses (8m) jul / aug04 6. For any two sets A and B, prove the following (4m) jul 07 A (A B) = A B 7. Determine the sets A and B given that A B = {1, 2, 4}, B A = {7, 8} and AUB = {1, 2, 4, 5, 7, 8, 9} (4m) jul Let M, P and C be the sets of students taking Mathematics courses, Physics courses and Computer Science courses respectively in a university. Assume M = 300, P = 350, C = 450, M \ P = 100, M \ C = 150, P \ C = 75, M \ P \ C = 10. How many students are taking exactly one of those courses? (7m) Jan /feb For any three sets A,B and C prove that (A-B)-C =A (BUC) = (A-C) (B-C) (6m) Jul Explain the laws of set theory: (8m) dec 09 /jan Determine the sets A and B given that A B = {1, 3, 7, 11}, B A = {2, 6, 8} and A B = {4, 9} (5m) jul /aug Prove that: A B= (B 1 1 A) U (A B ) = (B-A) U (A-B). (4m) aug 03 Dept. of CSE, SJBIT 1

2 13. Using Venn diagram, prove the following property of the symmetric difference: A (B C) = (A B) C (4m) aug Thirty cars are assembled in a factory. The options available are a transistor, an air conditioner and power windows. It is known that 15 of the cars have transistor, 8 of them have conditioners and 6 of them have power windows. Moreover, 3 of them have all three options. Determine at least how many cars do not have any options at all. (5m) Jan / feb A survey on a sample of 25 new cars showed that the cars had the following a. 15 cars had air conditioners b. 12 cars had radios c. 11 cars had power windows d. 5 cars had air conditioners and power windows e. 9 cars had air conditioners and radios f. 4 cars had radios and power windows g. 3 cars had all the three options h. Find the number of cars that had i) only power windows ii) at least one option (7m) Jan / feb A survey of 500 television viewers of sports channel produced the following information: 285 watch cricket, 195 watch hockey, 115 watch foot ball, 45 watch cricket and foot ball, 70 watch cricket and hockey, 50 watch hockey and foot ball and 50 do not watch any of the three kinds of games i) How many viewers in survey watch all three kinds of games? ii) How many viewers watch exactly one sport? (8m) Jul /aug The freshman class of a private engineering college has 300 students. It is known that 180 can program in PASCAL, 120 in FORTRAN, 30 in c++, 12 in PASCAL and c++, 18 in FORTRAN and c++, 12 in PASCAL and FORTRAN, and 6 in all three languages If two students are selected at random, what is the probability that they can i) Both program in PASCAL? ii) Both program only in PASCAL? (6m) Jan / feb A compuer services company has 300 programmers. It is known that 180 of these can program pascal, 120 in FORTRAN, 30 in c++, 12 in pascal and c++, 18 in FORTRAN and c++, 12 in pascal and FORTRAN and 6 in all the three. a. If a programmer is selected at random what is the probability that she can program in exactly two languages? b. If two programmers are selected at random what is the probability that they can both program in pascal? (10m) Jul /aug Define power set of a set. Obtain all the power sets of A={1,2,3,4} (3m) Jul 05 Dept. of CSE, SJBIT 2

3 UNIT 2: Fundamentals of logic 1. Discuss the basic connectives that are used in logic. (6m) May/Jun Define converse,inverse and contra positive of a statement: (6m) Jun Given p and q statements, explain the following terms a)conjunction b) disjunction c) logically Equivalence d) tautology (8M) May/Jun S how that (p v q) (q v p) is a tautology. (5m) jul Find the truth value of p,q,r for the following using truth tables: (5m) jul Prove the following tautologies: (5m) dec 09 / jan Prove the following: (6m) dec 09 / jan Find the truth values for the following logical expressions: (4m) jul Write the truth table for the following: (8m) jul Simplify the following compound statements: (6m) aug Verify whether the following logical expressions are tautology or contradiction using truth tables: (6m) jan Prove the following logical statement is a tautology: (5m) jan Prove the following logical statement is a tautology: (5m) jan Prove the following logical statement is a tautology: (5m) jan Prove the following logical statement is a tautology 5m) jan10 UNIT 3: Fundamental logic 1. Find inverse, converse and contra positive of the following: (6m) (may/jun 12) (dec 9 /jan10) 2. Simplify the following with reasons: (6m) may/jun Prove the following primitive statements: (8m) may/jun Prove below open statements: (5m) aug Prove below quantifiers: (6m) dec 9 6. Write inverse, converse and contra-positive: (5m) jul Prove the below open statements: (7m) jul Check the validity of the following arguments: (3m) jul Prove the following quantifiers: (3m) jul Prove the following rules of inferences: (4m) dec Prove the following rules of inferences (4m) dec prove the following rules of inferences (4m) jul Verify the rules of inference from the following truth tables: (6m) jul Prove the following open statements: (5m) jul Find converse inverse and contra positive of the logical expressions given below: (4m) jul 07 Dept. of CSE, SJBIT 3

4 UNIT 4: Properties of the integers 1. For n>=0 let fn denote the nth Fibonacci number. Prove that F0+f1+f2+.+fn= Summation Fi= fn+2-1 (4M)(may/jun 12) 2. Prove, by mathematical induction (6m) (may/jun 12) n (n + 2) = n (n +1) (2n + 7)/ 6 3. By induction prove that n! 2 n-1 forall integers n 1 (10m) (may/jun 12) 4. For all positive integers n, prove that if n>=24, then n can be written as a sum of 5s and 7s. (6m) dec Prove by induction: (6m) aug (2n-1) =n (2n-1) (2n+1) 3 6. Prove by mathematical induction. (5m) dec A sequence a n is defined by a1=3, an=an-1+an+1, for n>=2, find an explicit form: (6m) dec 07 UNIT 5: Relations and Functions 1. Let A = {1, 2, 3} Rand S be relations on A whose matrices are, (8m) ( may/jun 12) i) M R = and Ms = 000 Dept. of CSE, SJBIT 4

5 Determine relations R, R U S, R n Sand S -l and their matrix representation. 2. Define 1) reflexive 2) symmetric 3) Irreflexive 4) Anti symmetric 5) transitive relations: (6m)(may/jun 12) 3. A set of 3 members is (A, B, C). Brotherhood is the relation among them. Discuss whether the relation is equivalence. (6m) (may/jun 12) 4. Let A= {1,2,3,4,5}. Define a relation R on AXA by (x1,y1)r(x2,y2) if and only if x1+y1=x2+y2 (7m) jul Let A= {1,2,3,4,6} and r be the relation on A defined by(a,b) belongs to R if and only if a is a multiple of b. write down R as a set of ordered pairs. (4m) Dec Define a relation R on B as (a, b) R (c, d) if a + b = c + d. show that R is an equivalence relations. (6m) Dec 09 /jan A = {1, 2, 3} find a. R 1 = {(1, 1) (2, 2) (3, 3)} b. R 2 = {(1, 2) (2, 1) (1, 3) (3, 1) (2, 3), (3, 2)} c. c. R 3 = A x A (7m)dec If R is a relation on I, set of integers such that, x R y holds true if (x - y) is divisible by 5, show that R is an equivalence relation. (6m) Jul If R 1 and R 2 are equivalence relations defined on the same set A. prove that R 1 n R 2 is an equivalence relation. (7m) Aug 05 a. Given: R 1 and R 2 are equivalence relations. b. Prove: R 1 n R 2 is also equivalence relations. 10. Let A = {I, 2, 3, 4} and B = (A x A). Define a Relation R on B as (a, b) R (c, d) if a + b = c +d. S.T. R is an equivalence relation and compute B/R. (4m) jul Let, A = {a, b, c}, B = {1, 2, 3}, R = {(a, 1) (b, 1) (c, 2) (c, 3)} i) S = {(a, 1), (a, 2) (b, 1) (b, 2)} (6m) dec 09 Compute R ~, S ~, R U S, R n S, R -1, S -1 where(r ~ is R compliment) 12. Let A = {a, b, c} and Rand S be relations on A whose matrices are given below. Find the composite relation S o R, R o R, R o S, S o S and their matrices. (8m) Aug Let R = {(1, 2) (3, 4) (2, 2)} and S = {(4, 2) (2, 5) (3, 1) (1, 3)} be relations on the set A {2, 3, 4, 5} find S o R, R o S, Ro(S o R), So(R o S), Ro(R o R), So(R o R), S(S o S) (5m) dec If A= {1,2,3,4} B={2,5} C= {3,4,7} Determine: 1) AXB 2) BXA 3) AU (BXC) 4) (AUB)XC 5) (AXC)U(BXC) Define reflexive transitive and symmetric relations with respect to quantifiers. (5m)Aug Draw the hasse diagram for the poset (p(u)) where u={1,2,3,4} (4m)/Aug 04 Dept. of CSE, SJBIT 5

6 16. Let A={1,2,3,6,9,18} and define R on A by xry if x y. Draw hasse diagram of the poset. (4m) Aug Prove any R is a partial order. (4m) Jul 07 UNIT 6: Relations 1. Prove that 151 integers are selected from {1,2,3,.3000} then the selection must include two integers x,y where x y or y x. (4m) (may/jun 12) 2. Let f,g:z+->z+ where for all x belongs to Z+ f(x)=x+1 and g(x)=max{1,x-1} the maximum of 1 and x-1. State few properties. (10m) (may/jun 12) 3. Let f: Z->N be defined by f(x)= 2x-1 if x>0 and -2x for x<=0 (6m) (may/jun 12) Prove that f is one one and onto determine f-1 4. Find the nature of each of the function. (5m) Dec In each of the following cases sets A and B and a function f: A B are given. Determine (in each case) whether f is one to-one or onto (or both) (or neither) (5m) aug 03 i) 3. A = {I, 2, 3, 4} B = {a, b, c, d} ii) f = {(1, a), (2, a), (3, d), (4, c)} iii) 4. let A = {1,2,3,4} B = {a, b, c, d} f: A ( B is given as { (1, a), (2, b)} iv) f -1: B ( A = {( a, 1), (b, 2 )} v) f -1 is a function and hence f is invertible 6. let A,B,C be any three non-empty sets and A=B=C={set of real numbers} f: A ( B, g: f: B ( C be function defined by f(a) = a+1 and g(b) = b2 + 2, find a. gof (-2), b. fog (-2), c. gof(x), d. gog(x) (8m) aug Let X = {1, 2, 3} and f, g, h and s be function from X to x given by (8m) jul 07 f = {(1, 2), (2, 3), (3, 1)}, g = {(1, 2), (2, 1), (3, 3)}, h = {(1, 1), (2, 2), (3, 1)} s = {(1, 1), (2, 2), (3, 3)} 8. Let R denote the set of all real numbers. Let f: R ( R be a function defined by f(x) = x 2 Is f an invertible function? (5m) jul Find the inverse function f -1, of f: A B given by (6m) jul 08 A = B = {1, 2, 3, 4, 5} f = {(1, 3), (2, 2), (3, 4), (4, 5), (5, 1)} Dept. of CSE, SJBIT 6

7 10. Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Determine whether the following functions from A to B are invertible or not f = {(1, a), (2, a), (3, c), (4, d)} g = {(1, a), (2, c), (3, d), ( 4, d)} (8m) jul Prove that the symmetric difference is associative on sets. (5m) jul Suppose the addresses of customers of a bank are recorded in 101 files on the basis of hashing function.with the account number. As keys, determine the file in which the address of the customer with the account no is recorded (6m) dec Determine whether f(a,b)=[a+b) is commutative or assosiative. (5m) jul Let f,g,h:z->z be defined by f(x)=x-1 and g(x)=3x h(x)= 0 even and 1 odd. Determine the following: (5m) jan 06 UNIT 7: Groups 1. State and prove lagranges theorem. (10m) (may/jun 12) 2. Define abelian group. P T a group is abelian if and only if for all a,b belongs to G. (a,b) -1 =a -1 b -1 (4m) (may/jun 12) 3. Define a cyclic group. P T every cyclic group is abelian but converse is not true. (6m) (may/jun 12) 4. Let f:g->h be a homomorhism from G to H. If G is abelian P T H is also abelian. (3m) jul P T every subgroup is a cyclic group of itself. (5m) jun Define homomorhism and isomorhism. (5m) Dec09 7. Define the binary operation on o on Z by x o y=x+y+1. Verify that (Z,o) is an abelian group. (4m) Dec For any group G pt G is abelian if and only if (ab) 2 =a 2 b 2 (5m) dec If a group G P T (a -1 ) -1 =a and (ab) -1 =b -1 a -1 (4m) dec 08 jan P T for all groups (ab) -1 =a -1 b -1 (4m) 11. Find every subgroup of S5 for all 2<=n<=5. (5m) Jan Let g= S4 for a=[ ][ ] Find a subgroup and all the left cosets. (8m) Jun 09 UNIT 8: Group Codes 1. Define a rind and integral domain. (5m) (may/jun 12) 2. Let r be a commutative ring with unity. Prove that if and only if for all a,b,c belongs to r where a=/ 0 1b=ac=b=c. (6m) (may/jun 12) 3. Prove that every field is an integral (5m) (may/ju domain(6m)(may/jun 4. Show that Z 5 is an integral 12) domain. 5. Prove that Zn is a field if and inly if n is prime. (may/ju (5m) n 12) n 12) (4m) 6. Define ring with an exapmle: (4m) aug 07 Aug06 Aug08

8 7. Define hamming meric with example: (7m) aug Explain decoding with coset leaders. (7m) Jul Define ring with unity and ring with zero divisor: (5m) aug S is a sibring of R if and only if for all a,b belongs to S we have a+b belongs to S and ab belongs to S (5m) jul If R is a ring wiyh unity and a,b are units of R, P T ab is a unit of R and (ab) - 1 =b -1 a -1 (5m) aug Prove that a unit in aring R cannot be a proper divisor of zero. (4m) Jan If a is a unit of ring R P T a is also a unit of ring R. (2m) dec Let S and T be subrings of a ring R. P T S intersection T is a subring of R. (3m) (may/jun 12)

Discrete mathematical Structures. Unit State and prove De Morgan s law of set theory..(dec. 09/Jan 10, dec. 07/Jan 08)

Discrete mathematical Structures. Unit State and prove De Morgan s law of set theory..(dec. 09/Jan 10, dec. 07/Jan 08) Unit-1 1. State and prove De Morgan s law of set theory..(dec. 09/Jan 10, dec. 07/Jan 08) 2.. In a survey of 260 college students, the following data were obtained: 64 had taken a mathematics course, 94