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 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. How many of these students had taken none of the three courses? i. How many had taken only a computer science courses (dec. 07/Jan 08) 3. For any two sets A and B, prove the following (dec 11/Jan 12) 4.. 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} (dec 11/Jan 12) 5. 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? (jun/ July 2011) 6. For any three sets A,B and C prove that (A-B)-C =A (BUC) = (A-C) (B-C) (jun/ July 2011) 7. Explain the laws of set theory: ( May/ Jun 2010) 8. Determine the sets A and B given that A B = {1, 3, 7, 11}, B A = {2, 6, 8} and A B = {4, 9} (jun/ July 2011) 1 1 9. Prove that: A B= (B A ) U (A B ) = (B-A) U (A-B). (jul/aug 2011) 10. Using Venn diagram, prove the following property of the symmetric difference: (Dec.07/ Jan.08) 11. 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. (May/ Jun 2010) 12. A survey on a sample of 25 new cars showed that the cars had the following 15 cars had air conditioners 12 cars had radios 11 cars had power windows 5 cars had air conditioners and power windows Dept. Of ISE, SJBIT Page 1
9 cars had air conditioners and radios 4 cars had radios and power windows 3 cars had all the three options Find the number of cars that had i) only power windows ii) at least one option (Jun/July 2011) 13. 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? (Dec.08/ Jan.09) 14. 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? (Jun/Jul 08) 15. 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 thee survey sample of 120, what is the probability that they both want only iced tea with their meals? (Jun/Jul 08) 16. Find the probability of getting a sum different from 10 or 12 after rolling two dice. (Jun/ Jul 2011) 17. Explain set operations: (dec.07/ Jan 08) 18. 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? 1) If two programmers are selected at random what is the probability that they can both program in pascal? (jan / Feb 2006) b) 19. Define power set of a set. Obtain all the power sets of A={1,2,3,4}(May/ Jun 2010, Jul/ Aug 2003) 20. Simplify the following expression: (Dec 09/ Jan 10) Dept. Of ISE, SJBIT Page 2
Unit -2 1. Discuss the basic connectives that are used in logic (May / Jun 2010) 2. Given p and q statements, explain the following terms a) Conjunction b) disjunction c) logically Equivalence d) tautology (May / Jun 2011) 3.show that (p v q) (q v p) is a tautology. (Dec. 09 / Jan 2010) 4. Define converse,inverse and contra positive of a statement (Dec. 08 / Jan 09) 5. Find the truth value of p,q,r for the following using truth tables: (Dec.07/ Jan /08) 6. Proove the following tautologies: (Jan/ Feb 2006) 7. Proove the following: (Jan/ Feb 2006) 8. Find the truth values for the following logical expressions: (May/ Jun 2010) 9. Write the truth table for the following: ( Dec. 08/ Jan 09) 10. Simplify the following compound statements: ( Dec. 08/ Jan 09) 11. Verify whether the following logical expressions are tautology or contradiction using truth tables: (Jun/ Jul 08) 12. Prove the following logical statement is a tautology: (Jun/ Jul 08) Unit-3 1. Find inverse, converse and contra positive of the following: (Dec 08/ Jan 09) 2. Simplify the following with reasons: (Dec 08/ Jan 09) 3. Prrove the following primitive statements: (Jun / Jul 08) Dept. Of ISE, SJBIT Page 3
4. (jun/ Jul 11) 5. Write inverse, converse and contra-positive: (May/ Jun 2010) 6. Prove the below open statements: (Jul/ Aug 10) 7. Check the validity of the following arguments: (may/ Jun 2010) 8. Prove the following quantifiers: (Dec 09/ Jan. 10) 9. Prove the following rules of inferences: (Dec 07/ Jan 08).10. Prove the following open statements: (Dec 07/ Jan.08) Unit -4 1. For all positive integers n, prove that if n>=24, then n can be written as a sum of 5s and 7s. ( Jun/ Jul 2011) 2. Prove by induction: 1 2 + 3 2 +..+ (2n-1) 2 =n (2n-1) (2n+1) 3 ( Jun/ Jul 2011) 3. By induction prove that!n 2 n-1 for all integers n 1. ( Jan / Feb 20110) 4. Prove by induction: 1 2 +2 2 +3 2 +..+n 2 =n (2n+1) (n+1) 2 (Jun/ Jul 2011) 5. Prove, by mathematical induction 1.3 + 2.4 + 3.5 + n (n + 2) = n (n +1) (2n + 7) / 6(May/ Jun 2010) 6. A sequence an is defined by a1=3, an=an-1+an+1, for n>=2, find an explicit form: 7. For n>=0 let fn denote the nth Fibonacci number. Prove that F0+f1+f2+.+fn= Summation Fi= fn+2-1(may/ Jun 2010) Dept. Of ISE, SJBIT Page 4
Unit -5 1. 1. Let A= {1,2,3,4,5}. Define a relation R on AXA by (x1,y1)r(x2,y2) if and inly if x1+y1=x2+y2 (Dec 07/ Jan 08) 2. 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. (Dec. 07/ Jan 08) 3. Define 1) reflexive 2) symmetric 3) Irreflexive 4) Anti symmetric 5) transitive relations: ( Dec 06/ Jan 07) 4. A set of 3 members is (A, B, C). Brotherhood is the relation among them. Discuss whether the relation is equivalence. ( Dec 06/ Jan 07) 5. Define a relation R on B as (a, b) R (c, d) if a + b = c + d. show that R is an equivalence relations.( Jan/ Feb 2006) 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. R 3 = A x A 6. 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. ( Dec. 06/ Jan 07) 7. 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. ( Dec 06/ Jan 07) 8. 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. ( Jun/ July 08) 9. Let, A = {a, b, c}, B = {1, 2, 3}, R = {(a, 1) (b, 1) (c, 2) (c, 3)} S = {(a, 1), (a, 2) (b, 1) (b, 2)} Compute R ~, S ~, R U S, R n S, R -1, S -1 where(r ~ is R compliment) ( Jun/ July 08) ( Dec.08/ Jan 09) 10. Let A = {1, 2, 3} Rand S be relations on A whose matrices are, 11. 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., ( Jun/ July 08). 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).( Dec. 08/ Jan 09) 12. If A= {1,2,3,4} B={2,5} C= {3,4,7} Determine: ( Jun/ July 08) 13. 1) AXB 2) BXA 3) AU (BXC) 4) (AUB)XC 5) (AXC)U(BXC) ( Dec. 08/ Jan.09) 14. Define reflexive transitive and symmetric relations with respect to quantifiers. (Dec.07/ Jan.08) 15. Draw the hasse diagram for the poset (p(u)) where u={1,2,3,4} }( Jun/ Jul 2011) 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. }( Jun/ Jul 2011) 17. Prove any R is a partial order..( May/ Jun 2010) Dept. Of ISE, SJBIT Page 5
Unit -6 1. Find the nature of each of the function..(dec 09/ Jan 10) 2. 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). A =B= {1, 2, 3, 4} ( Dec.08/ Jan 09) 3.. let A = {1,2,3,4} B = {a, b, c, d} f: A ( B is given as { (1, a), (2, b)} f -1: B ( A = {( a, 1), (b, 2 )} ( Jun/ July 08) f -1 is a function and hence f is invertible 4. 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) ( Jun/ July 08) 5. Let X = {1, 2, 3} and f, g, h and s be function from X to x given by ( Jun/ July 08) 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)} 6. 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? (Dec.07/ Jan 08) 7. Find the inverse function f -1, of f: A B given by A = B = {1, 2, 3, 4, 5} ( Jun/ July 08) f = {(1, 3), (2, 2), (3, 4), (4, 5), (5, 1)} 8. 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 i. f = {(1, a), (2, a), (3, c), (4, d)} ii. g = {(1, a), (2, c), (3, d), ( 4, d)} ( Jun/ Jul 2011) 9.Prove that the symmetric difference is associative on sets. ( Dec.08/ Jan 09) Dept. Of ISE, SJBIT Page 6
10. 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. 2473876 is recorded ( Jun/ July 08) 11. Determine whether f(a,b)=[a+b) is commutative or assosiative.(dec.09/jan 10) 12. 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. ( Jun/ Jul 08) 13. 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..( Dec 08/ Jan 09) 14.. Let f: Z->N be defined by f(x)= 2x-1 if x>0 and -2x for x<=0 ( Dec.09/ Jan 10) 15. 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: (May/ June 2010) Unit- 7 1. State and prove lagranges theorem. (May / Jun 2010) 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 -1-1 ( May/ Jun 2010) b 3. Define a cyclic group. P T every cyclic group is abelian but converse is not true. Dec 09/ Jan 10) 4. P T every subgroup is a cyclic group of itself..( Dec 09/ Jan 10) 5. Define homomorhism and isomorhism.. ( Dec. 08/ Jan 09) 6. Define the binary operation on o on Z by x o y=x+y+1. Verify that (Z,o) is an abelian group. ( Jun/ July 08) 7. For any group G pt G is abelian if and only if (ab) 2 =a 2 b 2 ( Jun/ July 08) 8. If a group G P T (a -1 ) -1 =a and (ab) -1 =b -1 a -1 ( Dec 08/ Jan 09) 9. P T for all groups (ab) -1 =a -1 b -1 ( Jun/ July 08) 10. Find every subgroup of S5 for all 2<=n<=5. ( Jun/ July 08) 11. Let g= S4 for a=[ 12 3 4][2 3 4 1] Find a subgroup and all the left cosets. ( Dec 07/ Jan 08) Unit- 8 1. Define a rind and integral domain. ( jun/ Jul 2011) 2. Let r be a commutative ring with unity. Prove that if and only if for all a,b,c belons to r where a=/ 0 1b=ac=b=c. ( jun/ Jul 2011) 3. Prove that every field is an integral domain. ( Dec.08/ Jan 09) 4. Show that Z 5 is an integral domain. ( Dec 09/ Jan 10) 5. Prove that Zn is a field if and inly if n is prime.. ( Jun/ Jul 2011) 6. Define hamming meric with example: ( Dec 09/ Jan 10) 7. Explain decoding with coset leaders. ( Dec 08/ Jan 09) Dept. Of ISE, SJBIT Page 7
8. Define ring with unity and ring with zero divisor: ( Dec 08/ Jan 09) 9. 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( Dec 08/ Jan 09) 10. 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 ( Dec 08/ Jan 09) 11. Prove that a unit in aring R cannot be a proper divisor of zero. ( June/ Jul 08) 12.. If a is a unit of ring R P T a is also a unit of ring R. ( June/ Jul 08) 13. Let S and T be subrings of a ring R. P T S intersection T is a subring of R. ( Jun/ jul 10) Dept. Of ISE, SJBIT Page 8