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Question-1

Which of the following are sets?

        (i) The collection of all months of a year beginning with letter J.

        (ii) The collection of most talented writers of India.

        (iii) A team of eleven best cricket batsmen of the world.

        (iv) The collection of all boys in your class.

        (v) The collection of all natural numbers less than 100.

        (vi) The collection of novels written by the river Prem Chand.

        (vii) The collection of all even integers.

        (viii) The collection of different problems in this chapter.

        (ix) A collection of most dangerous animals of the world.


Solution:
(i), (iv), (v), (vi), (vii) and (viii) are sets.

Question-2

A  B = {1,2,3,4,5,6,7,8,9}
A B {1,2} and A = {1, 2, 3, 4, 5} find the set B.
Solution:
A B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A B = {1, 2} and

A = {1, 2, 3, 4, 5}

B = {1, 2, 6, 7, 8, 9}

Question-3

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, find the complements of the following sets:

        (i) A = {2, 4, 6, 8}

        (ii) B = {1, 3, 5, 7, 9}

        (iii) C = { 2, 3, 5, 7}

        (iv) φ

        (v) U


Solution:
(i) A' = {1, 3, 5, 7, 9}

(ii) B' = {2, 4, 6, 8}

(iii) C' = {1, 4, 6, 8, 9}

(iv) φ' = U

(v) U' = φ

Question-4

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol or in the blank spaces:

        (i) 5 ______ A

        (ii) 8 ______ A

        (iii) 0 ______ A

        (iv) 4 ______A

        (v) 2 ______ A

        (vi) 10 ______ A


Solution:
(i) 5 __ __ A

(ii) 8 __ __ A

(iii) 0 __ __ A

(iv) 4 __ __ A

(v) 2 __ __ A

(vi) 10 __ __ A

Question-5

If A B ={2, 3, 4, 5, 6, 8, 9, 11}, A B {5,8} and B = {2, 5, 8, 9} find the set A – B

Solution:
A B ={2, 3, 4, 5, 6, 8, 9, 11},

A B = {5,8}

B = {2, 5, 8, 9}

A = {3, 4, 5, 6, 8, 11}

A – B = {3, 4, 6, 11}

Question-6

If U is the set of all natural numbers and A' is the set of all composite numbers, what is A?

Solution:

U = {1, 2, 3, 4, 5, 6, ……………….}

A' = {4, 6,…………..}

Then A = {1, 2, 3, 5, …………..}

Question-7

Write the following sets in the roaster form:

        (i) A = {x : x is an integer and –3 x < 7}

        (ii) B = {x : x is a natural number less than 6}

        (iii) C = {x : x is two digit natural number such that sum of its digits is 8}

        (iv) D = {x : x is a prime number which is a divisor of 60}

        (v) E = the set of all letters in the word TRIGONOMETRY

        (vi) F = the set of all letters in the word SETS.


Solution:

(i) A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {1, 2, 3, 4, 5}

(iii) C = {17, 26, 35, 44, 53, 62, 71}

(iv) D = {2, 3, 5}

(v) E = {T, R, I, G, O, N, M, E, R, Y}

(vi) F = {S, E, T}

Question-8

If A = {p, q, r, s} find A A and A A.

Solution:
A = {p, q, r, s}

A A = {p, q, r, s}

A A = {p, q, r, s}

Question-9

Which of the following statements are true and which are false?

         (i) U' = φ

         (ii) φ ' = U

         (iii) For any two subsets, X and Y of U,
                (X Y)' = X' Y'

         (iv) For any two subsets, X and Y of U,
                 (X Y)' = X' Y'

         (v)  For any two subsets, S and T of U,
                (S T)' = S' T'

         (vi) For any two subsets S and T of U,
                (S T)' = S' T'


Solution:
(i) True

(ii) True

(iii) False

(iv) False

(v) True

(vi) True

Question-10

If A = {x : x is a letter in the word, ‘follow’} and
   B = {x : x is a letter in the word, ‘wolf’}, show that A = B.

Solution:
Clearly, A = {f, o , l, w} and B = {w, o, l, f}.

Since every element of A is in B and every element of B is in A, so A = B.

Question-11

Express the following sets by using the set builder method:

        (i) A = { 1, 3, 5, 7, 9}

        (ii) B = {2, 4, 6, 8}

        (iii) C = {-1, 1}

       (iv) D = {1, 5, 10, 15, ….}

        (v) E = {14, 21, 28, 35, 42, …..,98}


Solution:
(i) A = { x : x is an odd natural number, x  £ 9}

(ii) B = { x : x is an even natural number, x   8}

(iii) C= {x :  x is an odd natural number and |x| < 2}

(iv) D = {x : x is a natural number multiple of 5 and x = 1}

(v) E = {x : x is a multiple of 7 and 7<x<100}

Question-12

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 600, what is A?

Solution:
U = The set of all triangles in a plane.

A = The set of all triangles with at least one angle different from 600.

A' = The set of all equilateral triangles.

Question-13

List all the elements of the following sets:

         (i) A = {x : x is an odd natural number}  
    
         (ii) B = {x : x is an integer, -1/2<x<9/2} 

         (iii) C = {x : x is an integer, x24} 

         (iv) D = {x : x is a letter in the word "LOYAL"} 

         (v) E = {x : x is a month of a year not having 31 days}

         (vi) F = {x : x is a consonant in the English alphabet which precedes k}

Solution:
(i) A = {1, 3, 5, 7, 9, 11, 13….}

(ii) B = {0, 1, 2, 3, 4}

(iii) C = {-2, -1, 1, 2}

(iv) D = {L, O, Y, A}

(v) E = {February, April, June, September, November}

(vi) F = {b, c, d, f, g, h, j}

Question-14

Represent the following sets in a Venn diagram: U = {2, 3, 5, 7, 11}, A = {2, 3}

Solution:

Question-15

Match each of the sets on the left described in the roster form with the same set on the right described in set builder form:

         (i)   {1, 2, 3, 6}               (a) { x : x is a prime number and a divisor of 6 }

         (ii)  {2, 3}                        (b) {x : x is an odd natural number less than 10}

         (iii) {H, A, Y, R, N}           (c) {x : x is a natural number and divisor of 6.}

         (iv) {1, 3, 5, 7, 9}            (d) {x : x is a letter of the word 'HARYANA'.}

Solution:

(i) {1, 2, 3, 6}              (c) {x : x is a natural number and divisor of 6.} 

(ii){2, 3}                      (a) { x : x is a prime number and a divisor of 6 }

(iii) {H, A, Y, R, N}        (d) {x : x is a letter of the word 'HARYANA'.} 

(iv) {1, 3, 5, 7, 9}         (b) {x : x is an odd natural number less than 10}

Question-16

Which of the following sets is finite or infinite?

        (i) The set of the months of a year.

        (ii) {1, 2, 3, ……}

        (iii) {1, 2, 3, …….., 99, 100}

        (iv) The set of positive integers greater than 100.

        (v) The set of prime numbers less that 99.


Solution:
(i) Finite set

(ii) Infinite set

(iii) Finite set

(iv) Infinite set

(v) Finite set

Question-17

Represent the following sets in a Venn diagram:

        U = { x : x is a natural number and 2 x 8.}

        A = {x : x U and x divides 18}

        B = {x: x U and x is a prime divisor of 18}


Solution:
U = {2, 3, 4, 5, 6, 7, 8}

A = {2, 3, 6}

and B = {2, 3}


 

Question-18

Which of the following sets is finite or infinte?

         (i) The set of lines which are parallel to the x-axis.

         (ii) The set of letters in the English alphabet.

         (iii) The set of numbers which are multiples of 5.

         (iv) The set of animals living on earth.

         (v) The set of circles in plane passing through the origin.


Solution:

(i) Infinite set

(ii) Finite set

(iii) Infinite set

(iv) Finite set

(v) Infinite set

Question-19

If A, B and C are three subsets of the universal set U, draw a venn diagram showing A (B C)

Solution:
  

Question-20

Which of the following are examples of the null set?

        (i)    Set of odd natural numbers divisible by 2.

        (ii)   Set of even prime numbers.

        (iii) {x : x is a natural number, x<5 and simultaneously x>7}

        (iv) {y : y is a point common to any parallel lines}


Solution:
(i) Null set.

(ii) It is not a null set because 2 is a even prime number.

(iii) Null set.

(iv) Null set.

Question-21

If A, B and C are three subsets of the universal set U, draw a venn diagram showing (A B) C.

Solution:

Question-22

If A, B and C are three subsets of the universal set U, draw a venn diagram showing [(A B) C]'

Solution:

Question-23

In the following, state whether A = B or not:

          (i)    A = {a, b, c, d}                            B = {d, c, b, a}

          (ii)   A = {4, 8, 12, 16}                        B = {8, 4, 16, 18}

          (iii)  A = {2, 4, 6, 8, 10}                      B = {x :x is positive even integer less than 10}

          (iv)  A = {x : x is a multiple of 10}     B = {10, 15, 20, 25, 30,………….}

Solution:
(i) A = B

(ii) A B, because element 12 of set A is not present in set B and element 18 of set B is not present in set A. 

(iii) A = B

(iv) A B, because set B consists of elements that are multiple of 5.

Question-24

If A, B and C are three subsets of the universal set U, draw a venn diagram showing (A' B' ) C'

Solution:

Question-25

Are the following pair of sets equal? Give reasons.
          (i)  A = {2, 3} 
                B = {x: x is a solution of x2 + 5x + 6 = 0}

          (ii) A = {x:x is a letter in the word FOLLOW}
                B = {y:y is a letter in the word WOLF}

Solution:
(i) A = {2, 3}

    B = {x: x is a solution of x2 + 5x + 6 = 0} = {2, 3}

    Therefore the above pair are equal sets.

(ii) A = {x:x is a letter in the word FOLLOW} = {F, O, L, W}

     B = {y:y is a letter in the word WOLF} = {W, O, L, F}

     Therefore the above pair are equal sets.

Question-26

If A, B, and C are three subsets of the universal set U, draw Venn diagrams for the following:B C, when B C.

Solution:

Question-27

From the sets given below, select equal sets and equivalent sets

          A = {0, a}                          B = {1, 2, 3, 4}

          C = {4, 8, 12}                    D = {3, 1, 2, 4}

          E = {1, 0}                          F = {8, 4, 12}

          G = {1, 5, 7, 11}                H = {a, b}


Solution:
Equal sets:

(i) B = D

(ii) C = F;

Equivalent sets:

 (i) A, E, H;

 (ii) D, G;

Question-28

Given that A = {6, 7, 8, 9, 10} and B = {2, 3, 4, 5}. Write down all ordered pairs (a, b) such that a is divisible by b and hence write down the set ordered pairs given the relation ‘is a multiple of’ from A and B.

Solution:
A = {6, 7, 8, 9, 10} and B = {2, 3, 4, 5}

List of all ordered pairs : (6, 2), (6, 3), (6, 4), (6, 5), (7, 2), (7, 3), (7, 4), (7, 5), (8, 2), (8, 3), (8, 4), (8, 5), (9, 2), (9, 3), (9, 4), (9, 5), (10, 2), (10, 3), (10, 4), (10, 5)

The ordered pairs (a, b) such that a is divisible by b: (6, 2), (6, 3), (8, 2), (8, 4), (9, 3), (10, 2), (10,5)

Question-29

Which of the following statements are true?

          (i) The set of all cats is contained in the set of all animals.

          (ii) The set of all isosceles triangles is contained in the set of all equilateral triangles.

          (iii) The set of all rectangles is contained in the set of all squares.

          (iv) The sets A = {1} and B = {{1}} are equal.

          (v) The sets A = {x : x is a letter in the word "TITLE"} and B = {x : x is a letter in the word "LITTLE"} are equal.


Solution:

(i) True

(ii) False

(iii) False

(iv) False

(v) True

Question-30

Let R be the relation defined by ‘divides" from A = {2, 3, 5} to B = {6, 10, 12, 15}. Represent R (i) as a set of ordered pairs (ii) as a graph and (iii) by an arrow diagram.

Solution:

A = {2, 3, 5} to B = {6, 10, 12, 15}

(i) R = {(2, 6), (2, 10), (2, 12), (2, 15), (3, 6), (3, 10), (3, 12), (3, 15), (5, 6), (5, 10), (5, 12), (5, 15)}

(ii)

 

(iii)

 

Question-31

A and C are disjoint sets and both A and C are subsets of B, draw Venn diagrams for the following.

Solution:

Question-32

Make correct statements by filling in the symbols or in the blank spaces:

          (i)    {2, 3, 4} ……. {1, 2, 3, 4, 5,}

          (ii)   {a, b, c}……… {b, c, d}

          (iii)  {x : x is a student of Class 9 of your school.} ……..{x: x is a student of your school}
                                                                                                   
          (iv)  {x : x is a circle in the plane} ……..{x : x is a circle with radius 1}

          (v)   {x : x is a triangle in the plane} ………. { x : x is a rectangle in the plane}

          (vi)  {x : x is an equilateral triangle in the plane}……….{ x : x is a triangle in the plane}

          (vii) {x : x is an even natural number} ……… {x : x is an integer}

Solution:
(i)   {2, 3, 4} ……. {1, 2, 3, 4, 5,}

(ii)  {a, b, c}…
….. {b, c, d}

(iii) {x : x is a student of Class 9 of your school.} …
….{x; x is a student of your school.}

(iv) {x : x is a circle in the plane.} …
… {x : x is a circle with radius 1.}

(v)  {x : x is a triangle in the plane.} …
… { x : x is a rectangle in the plane.}

(vi) {x : x is an equilateral triangle in the plane .}…
…… { x : x is a triangle in the plane.}

(vii) {x : x is an even natural number.} …
… {x : x is an integer.}

Question-33

If A and B are two sets such that A has 21 elements, B has 17 elements, and A B has 21 elements, how many elements does A B have?

Solution:
We have n(A) = 12

             n(B) = 17

       n(A B) = 21

By using the formula,

       n(A B) = n(A) + n(B) - n(A B)

  We have 21 = 12 + 17 - n(A B)

     n(A B) = 29 -21

     n(A B) = 8

Question-34

If P = {2, 3, 4, 8, 9} write down the sets of ordered pairs representing the relations (i) is a factor of (ii) is divisible by (iii) is a multiple of 2 on P.

Solution:
P = {2, 3, 4, 8, 9}

(i) The sets of ordered pairs representing "is a factor of" : {(2, 2), (2, 4), (2, 8), (3, 3), (3, 9), (4, 4), (4, 8), (8, 8), (9, 9)}

(ii) The sets of ordered pairs representing "is divisible by" : {(2, 2), (4, 2), (8, 2), (4, 4), (8, 4), (8, 8), (9, 3), (9, 9), (3, 3)}

(iii) The sets of ordered pairs representing "is a multiple of 2 on P" : {((2, 2), (4, 2)), (8, 2)}

Question-35

Examine whether the following statements are true or false:

           (i)  {a, b} {b, c, a}

           (ii)  {a, e}
{ x : x is a vowel in the English alphabet.} 

           (iii) {1, 2, 3}
{1, 2, 3}

           (iv) {a}
{a, b, c} 

           (v)  {a}
{a, b, c} 

           (vi) {x : x is an even natural number less than 6.} 
{ x : x is a natural number which divides 36}

Solution:
(i) False, because elements a and b are present in that set.

(ii) True

(iii) True

(iv) True

(v) False

(vi) True

Question-36

Describe the relation R defined from A to B where A = {- 1, 2, 3, 4} to B = {-2, 4, 6} by the set R = {(- 1, - 2), (2, 4), (3, 6)}.

Solution:
A = {- 1, 2, 3, 4} to B = {-2, 4, 6}

set R = {(- 1, - 2), (2, 4), (3, 6)} is the relation "is half of"

Question-37

If A and B are disjoint sets, show that n(A B) = n(A) + n(B)

Solution:
We know that, if A and B are disjoint sets then n(A B) = φ .

Hence, by using the formula n(A B) = n(A) + n(B) - n(A B)

We have n(A B) = n(A) + n(B) - φ

              n(A B) = n(A) + n(B)

Example: Let A = {1, 2} and B = {3, 4},

      then A B = {1, 2, 3, 4} and A B = φ

        Now n(A) = 2, n(B) = 2, n(A B) = 4 and n(A B) = φ

Hence, n(A B) = n(A) + n(B)

Question-38

Let A = {1, 2, {3, 4}, 5}. Which of the following statements are false and why?

(i)     {3, 4} A

(ii)    {3, 4}
A

(iii)   {{3, 4}} A

(iv)   1
A

(v)    1 A

(vi)   {1, 2, 5}
A

(vii)  {1, 2, 5} A

(viii) {1, 2, 3}
A

(ix)   
φ A

(x)     {
φ } A

Solution:
(i)     False, {3,4} is an element not a set.

(ii)    True

(iii)   True

(iv)   True

(v)    False, 1 is an element not a set.

(vi)   True

(vii)  False, {1, 2, 5} is a set not an element.

(viii) False, 3 is an element of set contained in A.

(ix)   False, φ is not an element of A.

(x)    False, φ is not an element of A.

Question-39

Write the power set of A = {3,6,9}.

Solution:
P(A)= {φ , {3}, {9},{6},{3,6},{3,9},{6,9},{3,6,9}}.

Question-40

Describe the relation, domain and range if (i) R = {(1, 1), (8, 2), (27, 3), (64, 4)} (ii) R = {(Delhi, India), (Paris, France), (Karachi, Pakistan)} (iii) R = {(4, - 2), (9, - 3), (1, 1), (4, 2), (1, - 1), (9, 3)}

Solution:
(i) R = {(1, 1), (8, 2), (27, 3), (64, 4)}

R is the relation "is the cube of "

Domain = {1, 8, 27, 64}

Range = {1, 2, 3, 4}

(ii) R = {(Delhi, India), (Paris, France), (Karachi, Pakistan)}

R is the relation "is the capital of"

Domain = {Delhi, Paris, Karachi}

Range = {India , France, Pakistan}

(iii) R = {(4, - 2), (9, - 3), (1, 1), (4, 2), (1, - 1), (9, 3)}

R is the relation "is the square of"

Domain = {1, 4, 9}

Range = (- 3, - 2, - 1, 2 , 3}

Question-41

If x {1, 3, 7}, y {0, 2, 8} and R is the relation such that x + y < 8, represent R (i) as a set of ordered pairs and (ii) by an arrow diagram.

Solution:
(i) x {1, 3, 7}, y {0, 2, 8} , R is such that x + y < 8.

The set of ordered pairs = {(1, 0), (1, 2), (3, 0), (3, 2), (7, 0)}

(ii)

Question-42

Which of the following sets are equal ?

            A = {x : x N, x < 3},                          B = {1, 2},                       C = {3, 1}

            D = {x : x N, x is odd, x < 5},            E = {1, 2, 1},                   F = {1, 1, 3}

Solution:

A = {1, 2}, B = {1, 2}, C = {3, 1},

D = {1, 3}, E = {1, 2, 1}, F = {1, 1, 3}

A, B, E and C, D, F are equal sets.

Question-43

If A and B are two sets such that A B has 25 elements, A has 10 elements, and B has 37 elements, how many elements does A B have?

Solution:
n(A B) = 25; n(A) = 10 ; n(B) = 37

n(A B) = n(A) + n(B) – n(A B)

25 = 10 + 37 - n(A B)

n(A B) = 12

A B has 12 elements.

Question-44

In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea. Find :
          (i) how many drink tea and coffee both:
          (ii) how many drink coffee but not tea.

Solution:
Let A be the set of those persons who drink tea and let B be the set of those persons who drink coffee. Then,

A B = set of persons who drink both tea and coffee.

A – B = set of persons who drink tea but not coffee.

B – A = set of persons who drink coffee but not tea

n(A B) = 52, n(A-B) = 16 and n(A) = 33

Now, n(A-B) + n(A B) = n(A)

n(A B) = n(A) - n(A – B) = (33 – 16) = 17

Thus, 17 persons drink tea and coffee both.

Now, n(A) = 33, n(A B) = 52 and n(A B) = 17

n(A B) = n(A) + n(B) – n(A B)
n(A) = 33, n (A B) = 52 and n(A B) = 17.

n(A B) = n(A) + n(B) – n(A B)

n(B) = n(A B) + n(A B) – n(A)

n(B) = (52 + 17 – 33) =36

Also, n(B-A) + n(A B) = n(B)

n(B-A) = n(B) – n(A B) = (36 – 17) = 19

19 persons drink coffee but not tea.

Question-45

Represent the relation R from A = {2, 4, 5, 7} to B = {3, 5, 6, 8, 10} by an arrow diagram given a R b if b = a + 1 where a A and b B.

Solution:
A = {2, 4, 5, 7} to B = {3, 5, 6, 8, 10}

Question-46

Let A = {1, 2, 3, 4}, B = {1, 2, 3} and C = {2, 4}. Find all sets X satisfying each pair of conditions:

            (i) X B and X C

            (ii) X A, X B and X C

            (iii) X A, X B and X C


Solution:

(i) X = {1}, {3}, {1,2}, {1,3}, {2,3}, {1, 2, 3}

(ii) X = {1}, {3}, {1, 2}, {1, 3}, {2, 3}

(iii) φ , {2}

Question-47

Prove: (i) A – B A B and B A.

           (ii) A B and B C A C

Solution:
(i) A – B Every element of A is in B and every element of B is in A

                 A B and B A.

(ii) Let A B and B C.

     And, let x be an arbitrary element of A.

     Then x A x B [Since A B ]

     x C [Since B C ]

     A C

Question-48

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:

          (i)   1 A        

          (ii)  {1, 2, 3} A    

          (iii) {6, 7, 8} A   

          (iv) {{4, 5}} A  

          (v)  φ A    

          (vi) φ A

Solution:
(i) False

(ii) False

(iii) True

(iv) True

(v) False

(vi) True

Question-49

If relation R is defined on N by a R b if b = 3a + 1, is R a finite set?

Solution:
R = {(1, 4), (2, 7), (3, 10), (4, 13), …………}; R is a finite set.

Question-50

For any three sets A, B, C prove that: n (A B C) = n(A) + n(B) + n(C) + n(A B C) - [n(A B)] – [n(B C)]– [n(A C)]

Solution:
n (A B C) = n [(A B) C] = n(A B)+ n(C) - n [(A B) C]

                     = n(A B)+ n(C) - n [(A C) (B C)]

                     = n (A)+ n ( B) – n(A B)+ n(C) - n [(A C) + (B C) – n(A C B C) ]

                     = n(A) + n(B) + n(C) + n(A B C) -[n(A B)] – [n(B C)]– [n(A C)]

Question-51

How many elements has P(A) if A = φ ?

Solution:
P(A) = {φ }. Therefore P(A) has 1 element.

Question-52

If x R y implies y = 15 – 4x, given that x, y N, is R is finite set?

Solution:
The set R = {(1, 11), (2, 7), (3, 3)}; R is a finite set.

Question-53

          (ix) {{φ }} A

 


Solution:

(i) True

(ii) True

(iii) False, 1 is an element of A and not a set.

(iv) True

(v) False, 7 is an element of A and not a set.

(vi) True

(vii) True

(viii) True

(ix) True

Question-54

Examine what types of relations are the following (i) ‘is of equal length’ on the set of all line segments in a plane. (ii) ‘is congruent to’ on the set of triangles in a plane (iii) ‘is a son of’ on the set of family members.

Solution:
(i) ‘is of equal length’ on the set of all line segments in a plane is a reflexive, symmetric and transitive relation.

(ii) ‘is congruent to’ on the set of triangles in a plane is a reflexive, symmetric and transitive relation.

(iii) ‘is a son of’ on the set of family members is not reflexive, not symmetric and not transitive relation.

Question-55

If X = {8n – 7n - 1; n e} and Y = {49(n - 1); n N, then prove that x Y.

Solution:
Let xn = 8n – 7n - 1 = (1 + 7)n – 7n - 1 =

for n 2

For x = 1, xn = 0

Thus, X contains all positive integral multiplies of 49 of the form 40kn,

where kn = .

Also, y contains all positive integral multiplies of 49 including zero.

Thus X Y.

Question-56

Let A, B and C be three sets. If A B and B C, is it true that A C? If not give an example.

Solution:
No, A = {1, 2}, B = {1, 2, 3}, C = {{1, 2, 3}}

Question-57

Check which of the above relations is an equivalence relation.

Solution:
(i) ‘is of equal length’ on the set of all line segments in a plane is an equivalence relation.

(ii) ‘is congruent to’ on the set of triangles in a plane is an equivalence relation.

(iii) ‘is a son of’ on the set of family members is not an equivalence relation.

Question-58

For the following sets find their union:

             (i)   A = {a,e,i,o,u}; B = {a,b,c}.

          (ii)  X = {1,3,5}; Y = {1,2,3}.

          (iii) A = {x : x is a natural number and multiple of 3.};
                 B = {x : x is a natural number less than 6.}.

          (iv) A = {x : x is a natural number and 1 < x < 6.};
                 B = {x : x is a natural number and 6 < x < 10.}.

          (v)  A = {1,2,3}; B =
φ.

Solution:

i)   A U B = {a,e,i,o,u,b,c} 

ii)  X U Y = {1,2,3,5}

iii) A = {3,6,9,12,………}

     B = {1,2,3,4,5}

     A U B = { 1,2,3,4,5,6,9,12.................}

iv) A = {2,3,4,5,6}
     
     B = {7,8,9}

     A U B = {2,3,4,5,6,7,8,9}

     A U B = {x : 1 < x < 10, x
N}

v)  A U B = {1,2,3}

Question-59

Which of the following are reflexive? (a) ‘is similar to’ on the set of all triangles in a plane (b) ‘is relatively prime to’ on the set N (c) ‘is parallel to’ on the set of all lines on a plane (d) ‘is less than’ on the set N.

Solution:
(a) The relation ‘is similar to’ on the set of all triangles in a plane is reflexive.

(b) The relation ‘is relatively prime to on the set N is not reflexive.

(c) The relation ‘is parallel to’ on the set of all lines on a plane is reflexive.

(d) The relation ‘is less than’ on the set N not reflexive

Question-60

Let A = {a,b}, B = {a,b,c}. Is A What is A U B ?

Solution:
Yes, A U B = {a,b,c}

Question-61

Which of the following relations are symmetric? (a) ‘is a sister’ on the set of all members of a family (b) ‘is a multiple of’ on the set N (c) ‘is a divisor of the set of all integers (d) ‘is perpendicular to’ on the set of all lines of a plane.

Solution:
(a) The relation ‘is a sister’ on the set of all members of a family is not symmetric.

(b) The relation ‘is a multiple of’ on the set N is not symmetric.

(c) The relation ‘is a divisor’ on the set of all integers is not symmetric.

(d) The relation ‘is perpendicular to’ on the set of all lines on a plane is symmetric.

Question-62

If A = {2x/x N } and B = {2x+1 / x N} and subsets of the universal set X = N, find (i) A B (ii) A B (iii) A’ (iv) B’.

Solution:
A = {2, 4, 6, 8……}

B = {3, 5, 7, 9,….}

(i) A B = {x/ x N, x 1}

(ii) A B = ϕ

(iii) A’ = {2x – 1 / x N}

(iv) B’ = A {1}

Question-63

If A and B are two sets such that A then what is A U B ?

Solution:
A U B = B.

Question-64

Which of the following relations are transitive? (a) ‘is a friend of’ on the set of all students in a class (b) ‘is congruent to’ on the set of all triangles on a plane (c) ‘is married to’ on the set of all human beings (d) ‘is relatively prime to’ on the set N.

Solution:
(a) The relation ‘is a friend of’ on the set of all students in a class is not transitive.

(b) The relation ‘is congruent to’ on the set of all triangles on a plane is transitive.

(c) The relation ‘is married to’ on the set of all human beings is not transitive.

(d) The relation ‘is relatively prime to’ on the set N is not transitive.

Question-65

What type of relation R defined by (a) R = {(2, 2), (3, 2), (2, 3), (3, 3)} on A = {2, 3, 4} (b) R = {(- 1, 2), (2, - 1), (- 1, - 1), (2, 2), (3, 3)} on A = {- 1, 2, 3} (c) R = {(2, 3), (3, 3) (5, 5), (3, 2), (5, 7), (7, 5)} on A = {2, 3, 5, 7, 9}.

Solution:
(a) The relation R = {(2, 2), (3, 2), (2, 3), (3, 3)} on A = {2, 3, 4} is symmetric and transitive.

(b) The relation R = {(- 1, 2), (2, - 1), (- 1, - 1), (2, 2), (3, 3)} on A = {- 1, 2, 3} is reflexive and transitive.

(c) The relation R = {(2, 3), (3, 3) (5, 5), (3, 2), (5, 7), (7, 5)} on A = {2, 3, 5, 7, 9} is only symmetric.

Question-66

For the following sets find their intersection :

          (i)   A = {a,e,i,o,u} ; B = {a,b}

          (ii)  X = {1,3,5} ; Y = {1,2,3}

          (iii) A = {x:x is a natural number and multiple of 3.}
                 B = {x:x is a natural number less than 6.}

Solution:

i)  A B = {a}

ii)  XY = {1,3}

iii) A = {3,6,9,12,………}
     B = {1,2,3,4,5}

    ... A B ={3}

Question-67

Prove that A - (A-B) = A B.

Solution:

Let x A - (A-B)

x A and x (A – B)

x A and x A and x B

x A and x B

x A B

A – (A - B) A B ------------------- (i)

Let x A B

x A and x B

x A and x (A – B)

x A – (A-B)

   A B A – (A-B) ………………………………(ii)

From (i) and (ii)

   A – (A - B) = A B.

Question-68

Which of the following pairs of sets are disjoint?

          (i) {1,2,3,4} and {x: x is a natural number and 4
x 6.}

          (ii) {a,e,i,o,u} and {c,d,e,f}

          (iii) {x:x is an even integer.} and {x:x is an odd integer.}

Solution:
The pairs of sets in (iii) are disjoint.

Question-69

Check whether the following set defines an equivalence relation on the given set (a) R = {(p, p), (q, q), (r, r), (s, s), (p, q), (q, p), (s, r), (r, s)} on A = {p, q, r, s} (b) R = {(1, 3), (1, 1), (3, 1), (2, 3), (3, 3), (3, 2), (1, 2), (2, 1), (2, 2)} on A = {1, 2, 3}

Solution:
(a) The relation R = {(p, p), (q, q), (r, r), (s, s), (p, q), (q, p), (s, r), (r, s)} on A = {p, q, r, s} is an equivalence relation.

(b) The relation R = {(1, 3), (1, 1), (3, 1), (2, 3), (3, 3), (3, 2), (1, 2), (2, 1), (2, 2)} on A = {1, 2, 3} is an equivalence relation.

Question-70

Given the set A = {5, 7, 9} write down a set of ordered pairs on A which defines an equivalence relation on A.

Solution:
A = {5, 7, 9}

The set of all pairs of A = {(5, 5), (7, 7), (9, 9), (5, 7), (7, 5), (5, 9), (9, 5), (7, 9), (9, 7)}

Question-71

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

           (i)     A B

          (ii)    B C

          (iii)   A C B

          (iv)   A C

           (v)    B D

           (vi)   A (B C)

           (vii)  A D

           (viii) A (B D)

           (ix)   (A B) (B C)

           (x)    (A B) (B C)


Solution:

(i)     A B = {7, 9, 11}

(ii)    B C = {11, 13}

(iii)   A C B = {11}

(iv)   A C = {11}

(v)    B D = {}

(vi)   A (B C) = {11}

(vii)  A D = {}

(viii) A (B D) = {3, 5, 7, 9, 11} {} = {}

(ix)   (A B) (B C) = {7, 9, 11} {7, 9, 11, 13, 15} = {7, 9, 11}

(x)    (A B) (B C) = {3, 5, 7, 9, 11, 13} {7, 9, 11, 13, 15}

                                   = {7, 9, 11, 13}

Question-72

In the set N of natural numbers, examine the kinds of relation given below (i) If a2 + b2 is a perfect square, a R b (ii) If a2 = b, a R b.

Solution:
(i) If a2 + b2 is a perfect square, a R b

(3, 4), (4, 3), (5, 12), (12, 5), (8, 6), (6, 8),

R is symmetric relation.

(ii) If a2 = b, a R b is not a relation.

Question-73

Let A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number }, D = {x : x is a prime number}. Find

           (i) A B

           (ii) A C

           (iii) A D

           (iv) B C

           (v) B D

           (vi) C D


Solution:
A = {x : x is a natural number} = {1, 2, 3, 4, …..}

B = {x : x is an even natural number} = {2, 4, 6, 8, …….}

C = {x : x is an odd natural number } = {1, 3, 5, 7, 9, …….}

D = {x : x is a prime number} = {2, 3, 5, 7, ………}

(i) A B = {2, 4, 6, 8, …….} = B

(ii) A C = {1, 3, 5, 7, 9, …….} = C

(iii) A D = {2, 3, 5, 7, ………} = D

(iv) B C = {}

(v) B D = {2}

(vi) C D = {3, 5, 7, ………} = {x : x is an odd prime number}

Question-74

(d) f = {(1, 2), (4, - 3), (9, 4), (16, 1)}

 


Solution:
(a) A = {1, 4, 9, 16} to B = {- 1, 2, - 3, - 4}

f = {(1, - 1), (4, 2), (9, - 3), (16, - 4)} has one to one correspondence from the elements of A to B, f is a function from A to B.

(b) f = {(1, - 4) (1, - 1), (9, - 3), (16, 2)} does not have one to one correspondence from the elements of A to B, f is not a function from A to B.

(c) f = {(4, 2), (1, 2), (9, 2), (16, 2)} has one to one correspondence from the elements of A to B, f is a function from A to B.

(d) f = {(1, 2), (4, - 3), (9, 4), (16, - 1)} is a constant function, as every elements of A has same image in B, f is a function from A to B.

Question-75

Let A = {3, 6, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}. Find

          (i)     A – B

          (ii)    A – C

          (iii)   A – D

          (iv)   B – A

          (v)    C – A

          (vi)   D – A

          (vii)  B – C

          (viii) B – D

          (ix)   C – B

          (x)    D – B

          (xi)   C – D

          (xii)  D – C


Solution:
A = {3, 6, 12, 15, 18, 21},

B = {4, 8, 12, 16, 20},

C = {2, 4, 6, 8, 10, 12, 14, 16}  

and D = {5, 10, 15, 20}.

(i) A – B = {3, 6, 15, 18, 21}

(ii) A – C = {3, 15, 18, 21}

(iii) A – D = {3, 6, 12, 18, 21}

(iv) B – A = {4, 8, 16, 20}

(v) C – A = {2, 4, 8, 10, 14, 16}

(vi) D – A = {5, 10, 20}

(vii) B – C = {20}

(viii) B – D = {4, 8, 12, 16}

(ix) C – B = {2, 6, 10, 14}

(x) D – B = {5, 10, 15}

(xi) C – D = {2, 4, 6, 8, 12, 14, 16}

(xii) D – C = {5, 15, 20}

Question-76

X = {- 4, - 2, 0, 2, 4}, Y = {0, 1, 4, 9, 16} and f: X ###ERROR###à Y is defined by f(x) = x2. Check whether f is a function.

Solution:
f(x) = x2

f(-4) = (- 4)2 = 16

f(-2) = (- 2)2 = 4

f(0) = (0)2 = 0

f(2) = (2)2 = 4

f(4) = (4)2 = 16

Since 16, 4, 0, 4 and 16 are the elements of Y, f: X ###ERROR###à Y is a function.

Question-77

If X = {a, b, c, d} and Y = {f, b, d, g}, find (i) X – Y (ii) Y – X (iii) X Y

Solution:
(i) X – Y = {a, c}

(ii) Y – X = {f, g}

(iii) X Y = {b, d}

Question-78

Is f: N ###ERROR###à N defined by f(x) = (x + 1) a function? Justify your answer.

Solution:
f(x) = (x + 1)

f(1) = (1 + 1)

      = (2)

      = 1

f(2) = (2 + 1)

      = (3)

      = 1.5

When x = 2, f(2) = 1.5 is not an element of N since it is not a whole number.

f(x) = (x + 1) is not a function.

Question-79

If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q ?

Solution:
R – Q is the set of all irrational numbers.

Question-80

Find the range of the function f: A ###ERROR###à B where A = {- 3, - 2, - 1, 0, 1, 2}, B = {- 7 , - 4, - 3, - 1, 0, 2, 5, 8} given f(x) = 3x + 2.

Solution:
f(x) = 3x + 2

f(- 3) = 3(- 3) + 2 = - 9 + 2 = - 7

f(- 2) = 3(- 2) + 2 = - 6 + 2 = - 4

f(- 1) = 3(- 1) + 2 = - 3 + 2 = - 1

f(0) = 3(0) + 2 = 0 + 2 = 2

f(1) = 3(1) + 2 = 3 + 2 = 5

f(2) = 3(2) + 2 = 6 + 2 = 8

The range of the function f: A ###ERROR###à B is {- 7, - 4, - 1, 2, 5, 8}

Question-81

State whether each of the following statements is true or false. Justify your answer.

          (i) {2, 3, 4, 5} and {3, 6} are disjoint sets

          (ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets

          (iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoints sets

          (iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.


Solution:
(i) False, 3 is an element of both the sets.

(ii) False, a is an element of both the sets.

(iii) True

(iv) True

Question-82

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. 
          
         Find    (i) A’

                  (ii) B’

                  (iii) (A C)’

                  (iv) (A B)’

                  (v) (A‘)’

                  (vi) (B - C)’


Solution:
(i) A’ = {5, 6, 7, 8, 9}

(ii) B’ = {1, 3, 5, 7, 9}

(iii) (A C) = {3, 4}

      (A C)’ = {1, 2, 5, 6, 7, 8, 9}


(iv) (A B) = {1, 2, 3, 4, 6, 8}

      (A B)’ = {5, 7, 9}

(v)  (A‘) = {5, 6, 7, 8, 9}

      (A‘)' = {1, 2, 3, 4}


(vi) (B - C) = {2, 8}

      (B - C)’ = {1, 3, 4, 5, 6, 7, 9}

Question-83

Find the images of – 1, - 2, 3, 4 of the function f: Z ###ERROR###à Z is defined by f(x) = 3x.

Solution:
f(x) = 3x

f(- 1) = 3(- 1) = - 3

f(- 2) = 3(- 2) = - 6

f(3) = 3(3) = 9

f(4) = 3(4) = 12

The images of the function f(x) = 3x are –3, - 6, 9, 12.

Question-84

If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:

          (i) A = {a, b, c}

          (ii) B = {d, e, f, g}

          (iii) C = {a, c, e, g}

          (iv) D = {f, g, h, a}


Solution:
(i) A’ = {d, e, f, g, h}

(ii) B’ = {a, b, c, h}

(iii) C’ = {b, d, f, h}

(iv) D’ = {b, c, d, e}

Question-85

If f: N ###ERROR###à N is defined by f(x) = x2 - 1 check whether f is a function. If f is not a function, find for what choice of the codomain, f will be a function.

Solution:
f(x) = x2 – 1

f(1) = (1)2 – 1 = 0

f(x) = x2 – 1 is not a function since 0 is not an element of N.

If the codomain is the set of whole numbers, then f will be a function.

Question-86

Taking the set of natural numbers as the universal set, write down the complements of the following sets

          (i) {x : x N and x is even}

          (ii) {x : x N and x is odd}

          (iii) {x : x N and x = 3n for some n N}

          (iv) {x : x is a prime number}

          (v) {x : x N and x is a perfect square}

          (vi) {x : x N and x is a perfect cube}

          (vii) {x : x N and x + 5 = 8}

          (viii) {x : x N and 2x + 5 = 9}

          (ix) {x:x N and x 7}

          (x) {x:x N and x is a divisible by 3 and 5}


Solution:
(i) {x:x is an odd natural number}

(ii) {x:x is an even natural number}

(iii) {x:x N and x is not a multiple of 3}

(iv) {x:x is a positive composite number and x = 1}

(v) {x:x N and x is not a perfect square}

(vi) {x:x N and x is not a perfect cube}

(vii) {x:x N and x 3}

(viii) {x:x N and x 2}

(ix) {1, 2, 3, 4, 5, 6}

(x) {x:x N and x is not divisible by 3 and 5}

Question-87

A = {0, 1, 2, 3, 4,}, B = {- 6, - 3, 0, 6}. Relation f: A ###ERROR###à B is such that f(x) = - 3x. Check whether it is a function.

Solution:
A = {0, 1, 2, 3, 4,}, B = {- 6, - 3, 0, 6}

f(x) = - 3x

f (0) = - 3 (0) = 0

f(1) = - 3(1) = - 3

f(2) = - 3(2) = - 6

f(3) = - 3(3) = - 9

f(4) = - 3(4) = - 12

The elements of B are –6, -3, 0 and 6 f: A ###ERROR###à B is such that f(x) = - 3x is not a function.

Question-88

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that

          (i) (A B)’ = A’ B’

          (ii) (A B)’ = A’ B’


Solution:
(i) A B = {2, 3, 4, 5, 6, 7, 8}

(A B)’ = {1, 9}

A’ B’ = {1, 3, 5, 7, 9} {1, 4, 6, 8, 9} = {1, 9}

Therefore (A B)’ = A’ B’

(ii) A B = {2}

(A B)’ = {1, 3, 4, 5, 6, 7, 8, 9}

A’ B’ = {1, 3, 5, 7, 9} {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}

Therefore (A B)’ = A’ B’

Question-89

{(3, 10), (- 3, 4), (1, 8), (0, 7), (- 5, 2)}. Is this relation set a function?

Solution:
A = {3, - 3, 1, 0, - 5} and B = {10, 4, 8, 7, 2}

This relation is a function.

Question-90

Show that (A B) – (A B) = (A - B) (B - A).

Solution:

L.H.S = (A B) – (A B) = (A B) (A B)’ = (A B) (A’ B’) = (A B A') (A B B’)
                                                                                                    = (B A’)
(A B’) 
R.H.S = (A - B) (B - A) = (A B’) (B A’) = (A B’) (B A’) 

Therefore L.H.S = R.H.S

                                                                     

Question-91

In the function f(x) = x2 – x + 7, the domain of f is {1, 3, - 3} find the range of f.

Solution:
The domain of f is {1, 3, - 3}

f(x) = x2 – x + 7

f(1) = (1)2 – 1 + 7 = 1 – 1 + 7 = 7

f(3) = (3)2 – 3 + 7 = 9 – 3 + 7 = 13

f(- 3) = (- 3)2 – (- 3) + 7 = 9 + 3 + 7 = 19

The range of the function = {7, 13, 19}

Question-92

Find the images of X = {125, 64, 1} if f: X ###ERROR###à Y is defined by f(x) = cube root of x where Y = {1, 2, 3, 4, 5}.

Solution:
X = {125, 64, 1}

Cube root of 125 = 5

Cube root of 64 = 4

Cube root of 1 = 1

The images are 5, 4, 1.

Question-93

Shade the following sets:

A’ (B C)


Solution:
    

Question-94

Shade the following sets:

A’ (C - B) in the Venn diagram given in figure below


Solution:
     

Question-95

Find the pre-image of 2 under the function f = {0, - 1), (3, 2), (5, 3), (7, 2)}.

Solution:
f = {(0, - 1), (3, 2), (5, 3), (7, 2)}

(3, 2) and (7, 2) are having the element 2.

The pre-images of 2 are 3 and 7.

Question-96

Is the function from N ###ERROR###à N given by f(x) = 2x + 1, onto? Write down the range.

Solution:
f(x) = 2x + 1

f(1) = 2(1) + 1 = 3

f(2) = 2(2) + 1 = 5

f(3) = 2(3) + 1 = 7

The range of the function f(x) = 2x + 1 is {3, 5, 7, ………}.

Question-97

If X = {4, 6, 8, 10}, Y = {3, 4, 5, 6, 7} and f: X ###ERROR###à Y is given by f(x) = + 1 represent f as (i) a set of ordered pairs (ii) a table.

Solution:
X = {4, 6, 8, 10}, Y = {3, 4, 5, 6, 7}

(i) f(x) = + 1

f(4) = + 1 = 3

f(6) = + 1 = 4

f(8) = + 1 = 5

f(10) = + 1 = 6

The ordered pairs of the function f: {(4, 3), (6, 4), (8, 5), (10, 6)}

(ii)

 

x

4

6

8

10

y

3

4

5

6

 

 

Question-98

If A = {0, 1, 2, 3}, B = {3, 7, 11, 15, 17}, f: A ###ERROR###à B is defined by f(x) =4x + 3, represent f as (i) the set of ordered pairs (ii) a table (iii) a graph (iv) an arrow diagram.

Solution:
(i) A = {0, 1, 2, 3}, B = {3, 7, 11, 15, 17}

The set of ordered pairs = {(0, 3), (1, 7), (2, 11), (3, 15)}

(ii)

 

x

0

1

2

3

y

3

7

11

15

 

(iii)

(iv)

 

Question-99

(a) write its domain and range. Represent it using (b) an arrow diagram (c) a table.

 


Solution:
(a) Domain = {- 1, - 3, - 5, - 4}

Range = {2, 1, 6, 3}

(b)

 

(c)

 

x

- 1

- 3

- 5

- 4

y

2

1

6

3

 

 

Question-100

Decide, among the following sets, which are the subsets of which

          A = {all real numbers satisfying x2 - 8x + 12 = 0}

          B = {2, 4, 6}

          C = {2, 4, 6, 8, ...}

          D = {6}


Solution:
A = {all real numbers satisfying x2 - 8x + 12 = 0} = {2, 6}

B = {2, 4, 6}

C = {2, 4, 6, 8, ...}

D = {6}.

A B; A C; B C; D A; D B; D C.

Question-101

In each of the following, determine whether the statement is true or false . If it is false, give an example.

          (i) If x A and A B, then x B.
          
          (ii) If A B and B C, then A C.

          (iii) If A B and B C, then A C
          
          (iv) If A B and B C, then A C.

          (v) If x A and A B, then x B.

          (vi) If A B and x B, then x A.


Solution:
(i) False
    Let A = {x, 1}, B = {{x, 1}, 2}
    Then x B.

(ii) False

     Let A = {1, 2}, B = {1, 2, 3}, C = {{1, 2, 3}, 5}
     Then {1, 2} C

(iii) True

(iv) False

       Let A = {1, 2}, B = {3, 4}, C = {1,2,3}
      Then A C

(v) False
     Let A = {x, 1}, B = {3, 4}

Then x B

(vi) True

Question-102

Let B be a subset of A and let P(A:B) = {X P(A)| X B }.

   (i) Let B = {a, b} and A = {a, b, c, d}. List all the members of the set P(A:B).

   (ii) Show that P(A: φ ) = P(A).


Solution:
(i) {a, b}, {a, b, c}, {a, b, d}, {a, b, c, d}
 

(ii) P(A: φ ) ={{a},{b},{c},{d},{a, b}, {b,c},{c,d},{a,d},{b,d},{a, b, c}, {a, b, d},{a, b, c, d}}

     P(A) ={{a},{b},{c},{d},{a, b}, {b,c},{c,d},{a,d},{b,d},{a, b, c}, {a, b, d}, {a, b, c, d}}

      P(A: φ ) = P(A)

Question-103

Show that for any sets A and B, A = (A B) (A - B) and A (B - A) = A B.

Solution:
A = A U

   = A (B B’)

   = (A B) (A B’)

   = (A B) (A - B)

A (B - A) = A (B A’)

   = (A B) (A A’)

   = (A B) U

   = A B

Question-104

Using properties of sets, prove that A (B A) = A

Solution:
A (B A) = (A B) (A A) = U A = A

Question-105

Using properties of sets, prove that A (A B) = A

Solution:
A (A B) = (A A) (A B) = A U = A




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