1.-SETS-RELATIONS.pdf

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SETS - RELATIONS

1. Which of the following is not correct?1) cA A if and only if A

2) cA A if and only if A X , where X is a universal set3) If A B A C , then B = C4) A = B is equivalent to A C B C and A C B C

2. If X and Y are two sets, cX Y X equals1) X 2) Y 3) 4) X Y

3. Let 1F be the set of all parallelograms, 2F the set of rectangles, 3F the set of rhombuses,

4F the set of squares and 5F the set of trapeziums in a plane then 1F is equal to

1) 2 3F F 2) 2 3 4 1F F F F 3) 3 4F F 4) 3 1F F

4. Let A 1, 2,3, 4 and B 2,3,4,5,6 , then A B is equal to

1) 2,3, 4 2) 1 3) 5,6 4) 1,5,6

5. The set cc c cA B C A B C C is equal to1) c cB C 2) A C 3) cB C 4) CC C

6. In a mathematics class, 20 children had forgotten their rulers and 17 had forgotten theirpencils, "Go and borrow them from someone at once", said the teacher, 24 children leftthe room, then how many children had forgotten both is1) 11 2) 12 3) 13 4) 14

7. If Cn A 23,n A B C 6 , C C Cn A C B 9,n A C B 3 then the valueof n A B C is1) 4 2) 5 3) 6 4) 7

8. If nX 8 7n 1/ n N and Y 49 n 1 / n N , then1) X Y 2) Y X 3) X Y 4) X Y

9. Let R be set of points inside a rectangle of sides a and b a, b 1 with two sides alongthe positive direction of x- axis and y-axis and C be the set of points inside a unit circlecentred at origin, then1) R x, y : 0 x a,0 y b 2) R x, y : 0 x a,0 y b

3) 2 2C x, y : x y 1 4) R C R 10. If X is a finite set. Let P(X) denote the set of all subsets of X and let n(X) denote the

number of elements in X. If for two finite subsets A, B, n(P(A)) = n(P(B))+15 then n(B)= ........., n(A)=...........1) 6, 2 2) 8, 4 3) 0, 4 4) 0, 1

11. In a city, three daily newspapers A, B, C are published. 42% of the people in that city

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read A, 51% read B and 68% read C. 30% read A and B; 28% read B and C; 36% read Aand C; 8% do not read any of the three newspapers. The percentage of persons who readall the three papers is1) 25% 2) 18% 3) 20% 4) 30%

12. Suppose that the number of elements in the set S is 105 and that S is split into n subsets11m+2 elements each. If m is an integer, then m is1) 1 2) 2 3) 3 4) 4

13. Let S = {1,2,3,4}. The total number of unordered pairs of disjoint subsets of S is equal

to

1) 25 2) 34 3) 42 4) 41

14. Universal set, 5 4 2: 6 11 0U x x x x and A = {x : x2 5x + 6 = 0}

B = {x : x2 3x + 6 = 0}

Then, 'A B is equal to (2006)

1) {1, 3} 2) {1, 2, 3} 3) {0, 1, 3} 4) {0, 1, 2, 3}

15. 3 22 1:4 3xx R R

x x x

equals

1) 0R 2) 0,1,3R 3) 0, 1, 3R 4 )10, 1, 3,2

R

16. Let R be the relation over the set of all straight lines in a plane such that 1 2 1 2l Rl l l .Then, R is1) symmetric 2) reflexive 3) transitive 4) an equivalence

17. If 2: 5 6 0A x x x , 2, 4B , 4,5C , then A B C is1) 2,4 , 3, 4 2) 4,2 , 4,33) 2,4 , 3, 4 , 4, 4 4) 2,2 , 3,3 , 4,4 , 5,5

18. If the relation R : A B , where A 1, 2,3 and B 1,3,5 is defined by

R x, y : x y, x A, y B , then1) R 1,3 , 1,5 , 2,3 , 2,5 , 3,5 2) R 1,1 , 1,5 , 2,3 , 3,5

3) 1R 3,1 , 5,1 , 3, 2 , 5,3 4) 1R 1,1 , 5,1 , 3,2 , 5,3 19. Let A 1, 2,3 and R 2,2 , 3,1 , 1,3 , then the relation R on A is

1) reflexive 2) symmetric 3) transitive 4) None of these20. If a relation R is defined on the set Z of integers as follows 2 2, 25a b R a b

Then, Domain (R) =

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1) 3, 4,5 2) 0,3, 4,5 3) 0, 3, 4, 5 4) 0, 5

21. Let R 2,3 , 3, 4 be relation defined on the set of natural numbers. The minimumnumber of ordered pairs required to be added in R so that enlarged relation becomes anequivalence relation is1) 3 2) 5 3) 7 4) 9

22. The minimum number of elemetns that must be added to the relation R 1, 2 , 2,3on the set of natural numbers so that it is an equivalence is1) 4 2) 7 3) 6 4) 5

23. Let N denote the set of all natural numbers and R a relation on N N . Which of thefollowing is an equivalence relation?1) a, b R c,d if ad b c bc a d 2) a, b R c,d if a d b c

3) a, b R c,d if ad bc 4) All the above24. If R is a relation from a finite set A having m elements to a finite set B having n elemetns,

then the number of relations from A to B is1) mn2 2) mn2 1 3) 2mn 4) nm

25. Let S be the set of all real numbers. Then the relation , :1 0R a b ab on S is1) Reflexive and symmetric but not transitive2) Reflexive and transitive but not symmetric3) Symmetric and transitive but not reflexive4) Equivalence relation

26. Which one of the following is an elementary symmetric function of 1 2 3 4, , ,x x x x

1) 1 2 3 2 3 4x x x x x x 2) 1 2 2 3 3 1x x x x x x

3) 2 2 2 21 2 3 4x x x x 4) 1 2 1 3 1 4 2 3 2 4 3 4x x x x x x x x x x x x 27. Let R be the real line. Consider the following subsets of the plane R R

, : 1S x y y x and 0

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R is1) a function 2) transitive 3) not symmetric 4) reflexive

Key & Hints

1. Ans :(3)cA satisfies (1) and (2) by definition, (4) also follows trivially..

Assuming A to be any set other than the empty set, also B = A and C , we haveA B A A C But B C , So (3) is incorrect

2. Ans :(3)According to De Morgan's Law, we have

c c cX Y X X Y X c cX X Y

3. Ans :(2)Since every rectangle, rhombus and square is a parallelogram so

1 2 3 4 1F F F F F 4. Ans :(4)

A BA B x : x or xB A

5. Ans :(3)

cc c cA B C A B C C = c cA B C A B C C

C CA A B C C C CB C C C = CB C .

6. Ans :(3)n(R)=20, n(P)=17 n R P 24

n R P 20 17 24 13 7. Ans :(4)

n A B C 23 6 9 3 5 8. Ans :(1)

We have ,n8 7n 1 n7 1 7n 1

n 2 n 3 n n2 3 nC 7 C 7 ... C 7

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n n n n 22 3 n49 C C 7 ... C 7 for n 2 49 Integer X Y

9. Ans :(2)Since, R denotes the set of points inside the rectangle of sides a and b for both a & b >1, then

R x, y : 0 x a,0 y b 10. Ans :(3)

Let n A m,n B n

m nn P A 2 ,n P B 2

n P A n P B 15 m n2 2 15 m 4;n 0

11. Ans :(1)Let the no. of persons in the city be 100.Then we have

n A 42, n B 51, n C 68 ;

n A B 30, n B C 28,n A C 36

n A B C 100 8 92

Using n A B C n A n B n A B

n B C n A C n A B C Substituding the above values, we have

92 42 51 68 30 28 36 n A B C

n A B C 92 161 94

n A B C 92 67 25 12. Ans :(3)

Number of elements in each set = 11m+2Number of elements in n subsets = n(11m+2)

n 11m 2 105 21 5 3 7 5 n 3 11m 2 35 m 3

13. Ans: (4)Required number

43 1 412

14. Ans: (3)

5 4 3 2: 6 11 6 0 0,1,2,3U x x x x x

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2: 5 6 0 2,3A x x x and 2: 3 2 0 2,1B x x x

'A B U A B

0,1,2,3 2 0,1,3 15. Ans: (3)

Let 3 22 1:4 3xA x R

x x x

Now, 3 2 24 3 4 3x x x x x x 3 1x x x

0, 1, 3A R

16. Ans: (1)Clearly, R is not a reflexive relation, because a line cannot be perpendicular to itself.Let 1 2l Rl , Then

1 2 1 2l Rl l l

2 1l l

2 1l Rl R is a symmetric relation.R is not a transitive relation, because if

1 2l l and 2 3l l , then 1l may be parallel to 3l .17. Ans: (1)

We have, 2,3A , 2, 4B and 4,5C

4B C

2, 4 , 3, 4A B C 18. Ans: (1)

Since, R x, y : x y, x A, y B i.e. R contains all those pairs for which the first number from the set A is less than thesecond from the set B.

19. Ans: (2)Not ReflexiveAs all like pairs are not there.Symmetric but not TransitiveAs (3, 1) and (1, 3) R but (3, 3) R 26. We have 2 2, 25a b R a b

225b a Clearly, 0 5a b

3 4a b 4 3a b

and 5 0a b

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20. Ans: (3)We have 2 2, 25a b R a b

225b a Clearly, 0 5a b

3 4a b 4 3a b

and 5 0a b 21. Ans: (4)

To make it reflexive, we need to add (2, 2), (3, 3), (4, 4). To make symmetric it requires(3, 2), (4, 3) to be added. to make transitive (2, 4) and (4, 2) must be added so, therelationR = {(2, 2), (3, 3), (4, 4), (2, 3), (3, 2), (3, 4), (4, 3), (2, 4), (4, 2)}

22. Ans: (2)Since R 1, 2 , 2,3i.e. A 1, 2 and B 2,3Now if R1 is the Reflexive relation, such that

1R 1,2 , 2,3 , 1,1 , 2, 2 , 3,3has 5 elementsNow, If 2R is both symmetric & reflexive relation, then

2R {(1, 2), (2, 3), (1, 1), (2, 2),(2, 1), (3, 2), (3, 3)}

has 7 elementsAgain, 3R is reflexive, symmetric and transitive all together, then

3

1, 2 2,3 1,1R 2, 2 2,1 3, 2

3,3 1,3 3,1

has 9 elemtns. Starting from 2 elements, therfore the minimum number of elements tobe added is 7.

23. Ans: (4)The relation in (A) is reflexive because ab = ba and a b b a , so that

ab b a ba a b . i.e. a, b R a,b .It is also symmetric because ad b c bc a d , i.e. a, b R c,d , implies

cb d a da c b , i.e. c,d R a, b . Simple computations show that relation (A) isTRANSITIVE, too, and that the relations in (B) and (C) are also reflexive, symmetricand transitive.

24. Ans: (1)A B will have mn ordered pairs. Each subset of A B will be a relation. The numberof subsets of a set consisting of mn elements will be mn2 .

25. Ans: (1)Clearly R is reflexive and symmetricTransitivity: We observe that 1,1/ 2 R and 1/ 2, 1 R but 1, 1 R because

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1 1 1 0 , not > 0.So, R is not transitive on R.

26. Ans: (4)Numbers of elements of 1 2 3 4, , ,x x x x A (say) is

4n A

Number of elementary functions 4 2 6C and series sets are given in (4)

27. Ans: (2)

, ; 1,0 2S x y y x x S is not symmetric , : intT x y x y is an eger clearly T is an equilence

28. Ans: (1)Let x, x have every letter common.

,x x R . Thus R is reflexive

Let ,x y R thus x, y have atleast one letter is common.y, x have atleast one letter in commonThus R is symmetriccot x = AND, y = NEXT, Z= HERthen ,x y R and ,y z R But ,y z RThus R is not transitive

29. Ans: (4)

3,6,9,12A and 3,3 , 6,6 , 9,9 , 12,12 R R is reflexive

6,12 R but 12,6 R R is not symmetric

3,6 , 6,12 3,12R R R R is transitive

30. Ans: (3)

2, 4 , 2,3 2R has two images R is not a reflexive

1,1 R R is not reflexive

2,3 , 3,2R R R is not symmteric

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1.-SETS-RELATIONS.pdf