## Solutions to set-V

In this page, 'Solutions to set-V' we are discussing how to do the problems given in problems on set-V.

The following problems are under complement of a set.

1. (i) If U = {x: 0 ≤ x ≤ 10, x ∈ W} and A = {x: x is a multiple of 3}. Find A'.

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

Solution:

(i) First let us write the given sets.

U = {0,1,2,3,4,5,6,7,8,9,10}

A =  {3,6,9}

So A' is the set of all elements in U which are not in A.

A' =  {0,1,2,4,5,7,8,10}

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2.       If  U =   {a, b, c, d, e, f, g,h},  A= {a, b, c, d} and B = { b, d, f, g}, find

(i) AB      (ii) (A∪B)'       (iii)  A∩B         (iv) (A∩B)'

Solution:

(i)     AB     =   {a, b, c, d} ∪ { b, d, f, g}

=    {a,b, c, d,f,g}

(ii)     (A∪B)'   =    {e,h}

(iii)     A∩B      =    {b,d}

(iv)    (A∩B)'   =     {a,c,e,f,g,h}

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3.    If    U  =    { x: 1≤ x ≤ 10, xℕ},  A = {1,3, 5, 7, 9} and B = {2, 3,5, 9, 10},

find (i) A'     (ii) B'       (iii) A'∪B'      (iv) A'∩B'

Solution:

First we will write the given sets A and B.

U   =  {1,2,3,4,5,6,7,8,9,10}

A   =   {1,3,5,7,9}

B   =   {2,3,5,9,10}

(i)          A'  =   {2,4,6,8,10}

(ii)         B'  =    {1,4,6,7,8}

(iii)    A'∪B'  =      {2,4,6,8,10} ∪ {1,4,6,7,8}

=     {1,2,4,6,7,8,10}

(iv)   A'∩B'    =      {2,4,6,8,10}   {1,4,6,7,8}

=     {4,6,8}

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The following problems are under difference of sets.

4.   Given that U= {3,7,9,11,15,17,18}, M = { 3,7, 9, 11} and

N = { 7, 11, 15, 17},

find (i) M-N     (ii) N-M       (iii) N'-M     (iv)  M'-N

(v) M∩(M-N)       (vi)   N∪(N-M)    (vii)n(M-N)

Solution:

U =  {3,7,9,11,15,17,18}

M =  { 3,7, 9, 11},           M' = {15,17,18}

N = { 7, 11, 15, 17}       N'  = {3,9, 18}

(i)          M-N = {3,9}

(ii)             N-M  = {15,17}

(iii)            N'-M =  {18}

(iv)            M'-N   = {18}

(v)       M∩(M-N)  = { 3,7, 9, 11} ∩ {3,9}

=  {3, 9}

(vi)      N∪(N-M)  =  { 7, 11, 15, 17} ∪ {15,17}

=   {7, 11, 15, 17}

(vii)       n(M-N)   =    2.

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5.    If  A= {3, 6, 9, 12, 15, 18},  B = {4, 8, 12, 16, 20},

C = {2, 4, 6, 8, 10, 12}  and D = {5, 10, 15, 20, 25},

find  (i) A-B      (ii)  B-C    (iii) C-D      (iv) D-A

(v)  n(A-C)       (vi)   n(B-A)

Solution:

Given A= {3, 6, 9, 12, 15, 18},

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

C = {2, 4, 6, 8, 10, 12}

D = {5, 10, 15, 20, 25}

(i)          A-B     =    (Elements only in A not in B)

=    {3, 6, 9  15, 18}

(ii)         B-C     =    {16, 20}

(iii)        C-D     =    {2, 4, 6, 8,12}

(iv)        D-A     =    {5, 10, 20, 25}

(v)       n(A-C)   =   n({3, 9, 15, 18})

=        4.

(vi)      n(B-A)    =   n({4, 8, 16, 20})

=        4.

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6.    Let  U = {x: x is a positive integer less than 50},

A = {x: x is divisible by 4} and

B = {x: x leaves remainder 2 when divided by 14},

(i) List the elements of U, A and B

(ii) Find A∪B, A∩B, (A∪B)', n( A∩B), A-B and B-A

Solution:

First let us write the given sets.

(i)         U  =  {1,2,3,4,5,6,7.......49}

A  =  {4,8,12,16,20,24,28,32,36,40,44,48}

B  =   {16, 30,44}

(ii)     A∪B    =    {4,8,12,16,20,24,28,30, 32,36,40,44,48}

A∩B    =     {16, 44}

(A∪B)'  =    {1,2,3,5,6,7,9,10,11,13,14,15,17,18,19,21,22,23,25,26, 27,

29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47,49}

n(A∩B) =       2

A-B     =     {4,8,12,20,24, 28,32,36,40,48}

B-A     =      { 30}

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7.   Find the symmetric difference between the following sets.

(i)  X = {a, d, f, g,h},          Y = {b, e, g, h, k}

(ii)   P = {x: 3<x <9, xℕ},  Q = { x: x<5, x∈W}

(iii)  A = {-3, -2,0, 2, 3, 5},   B = {-4,-3,-1, 0, 2, 3}

Solution:

(i)     Symmetric difference between  X and Y is

X∆Y    =   (X-Y) ∪ (Y-X)

=   {a, d, f} ∪ {b, e, k}

=   {a, b, d, e,f,k}

(ii)       Symmetric difference between P and Q is

P∆Q   =   (P-Q) ∪ (Q-P)

=    [{4, 5, 6,7,8}- {0, 1, 2, 3, 4}] ∪ [{0,1,2,3,4}-{4,5,6,7,8}]

=    {5,6,7,8} ∪ {0,1,2,3}

=    {0,1,2,3,5,6,7,8}

(iii)      Symmetric difference of A and B is

A∆B   =    (A-B) ∪ (B-A)

=    {-2, 5} ∪ {-4, -1}

=    {-4, -2, -1, 5}

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8.    Use the Venn diagram, to answer the following questions.

(i) List the elements of  E, F, E∪F and E∩F

(ii) Find n(U), n(E∪F ) and n(E∩F)

Solution:

(i) Elements of E =  {1, 2, 4, 7}

F =   {4,7, 9,11}

E∪F =     {1, 2, 4, 7, 9, 11}

E∩F  =     {4,7}

(ii)                 n(U)  =     7

n (E∪F)  =     6

n((E∩F)  =      2

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9.    use the Venn diagram to answer the following questions.

(i) List U, G and H

(ii) Find G', H', G'∩H', n(G∪H)' and n(G∩H)'

Solution:

(i)   Elements of  U   =  {1,2,3,4,5,6,8,9,10}

G   =  {1,2,4,8}

H   =   {2,6,8,10}

(ii)                     G'  =    {3,5,6,9,10}

H'  =    {1,3,4,5,9}

G'∩H'   =      {3,5,9}

n(G∪H)'   =       3

n(G∩H)'   =       7

Parents and teachers can encourage the students to do the problems on their own and become master in union and intersection of sets.  If you have any doubt you can verify the solutions given in the above page 'Solutions to set-V'. Still if you have any doubt you can contact us through mail, and we will help you to clear all your doubts.