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Q. 11 Let A and B be sets. If A \cap X =B \cap X = \phiand A \cup X = B \cup X for some set X, show that A =B.

Given,  A  X B  X     and  A  X  B  X To prove:   A = B A = A (AX)              (A  X B  X)     = A (BX)      = (AB)  (AX)     =  (AB)              (A  X  )     =  (AB)  B = B (BX)              (A  X B  X)     = B (AX)      = (BA)  (BX)     =  (BA)              (B  X  )     =  (BA)  We know that    (AB) =  (BA) = A = B Hence, A = B        

Q10. Show that A \cap B = A \cap C need not imply B = C.

Let,       A = {0,1,2}              B = {1,2,3}              C = {1,2,3,4,5}  Given,   A  B = A  C L.H.S :     A  B = {1,2} R.H.S :    A  C = {1,2} and here   {1,2,3}  {1,2,3,4,5}  =   B  C. Hence,  A  B = A  C need not imply B = C.  

Q9. Using properties of sets, show that

(ii) A \cap ( A \cup B ) = A

This can be solved as follows (ii) A  ( A  B ) = A    A  ( A  B ) =  (A  A)  ( A  B )     A  ( A  B ) =  A  ( A  B )                          {  A  ( A  B ) = A  proved in 9(i)}    A  ( A  B ) =   A

Q. 12 Find sets A, B and C such that A \cap B, B \cap C and A \cap C are non-empty sets and A  \cap B \cap C = \phi

Given,    A  B, B  C and A  C are non-empty sets To prove : A   B  C   Let A = {1,2}       B = {1,3}       C = {3,2} Here,   A  B = {1}              B  C = {3}              A  C = {2} and   A   B  C                     

Q9. Using properties of sets, show that

(i) A \cup ( A \cap B ) = A

(i) A  ( A  B ) = A We know that  A  A             and       A  B  A           A  ( A  B )    A and also , A   A  ( A  B ) Hence,  A  ( A  B ) = A           

Q8. Show that for any sets A and B,

A = ( A \cap B ) \cup ( A – B ) and A \cup ( B – A ) = ( A \cup B )

A = ( A  B )  ( A – B ) L.H.S = A = Red coloured area R.H.S =  ( A  B )  ( A – B ) ( A  B ) = green coloured ( A – B ) =  yellow coloured ( A  B )  ( A – B ) = coloured part Hence, L.H.S = R.H.S = Coloured part   A  ( B – A ) = ( A  B ) A = sky blue coloured ( B – A )=pink coloured  L.H.S = A  ( B – A ) = sky blue coloured + pink coloured  R.H.S = ( A  B ) = brown coloured part L.H.S =...

Q. 13 In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?

n ( taking tea) = 150 n (taking coffee) = 225 n ( taking both ) = 100 n(people taking tea or coffee) = n ( taking tea) +  n (taking coffee) -  n ( taking both )                                                 = 150 + 225 - 100                                                 =375 - 100                                                  = 275 Total students = 600 n(students  taking neither tea nor...

Q. 14 In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?

n (hindi ) = 100 n (english ) = 50 n(both) = 25 n(students  in the group ) = n (hindi ) +  n (english ) - n(both)                                           = 100 + 50 - 25                                            = 125 Hence,there are 125 students in the group.

Q6. Assume that P ( A ) = P ( B ). Show that  A =  B 

Given,    P ( A ) = P ( B ) To prove :   A =  B  Let,    x A          A   P ( A ) = P ( B ) For some C  P ( B )  ,  x  C  Here, C  B  Therefore, x   B      and    A  B Similarly we can say B  A. Hence,   A =  B 

Q. 15 In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:

(i) the number of people who read at least one of the newspapers.

(ii) the number of people who read exactly one newspaper.

n(H) = 25 n(T) = 26 n(I) = 26 n(H  I) = 9 n( T  I ) = 8 n( H  T ) = 11 n(H  T  I ) = 3   the number of people who read at least one of the newspapers = n(HTI) = n(H) + n(T) + n(I) - n(H  I) -  n( T  I ) - n( H  T ) + n(H  T  I )                                                                                                      = 25 + 26 + 26 - 9 - 8 - 11 + 3                                    ...

Q5. Show that if A \subsetB, then C – B \subset C – A.

Given ,      A  B To prove :  C – B   C – A Let, x  C - B means    x C  but  xB A  B  so x C but   xA      i.e.   x  C - A Hence,    C – B   C – A

Q4. Show that the following four conditions are equivalent :

(i) A \subset B(ii) A – B = \phi (iii) A \cup B = B (iv) A \cap B = A

 

First, we need  to show  A B    A – B = Let  A  B  To prove : A – B = Suppose  A – B   this means, x  A and x  B , which is not possible as  A  B . SO,   A – B = . Hence, A  B  A – B = . Now, let A – B = To prove :  A  B  Suppose, x  A  A – B =  so x  B  Since,  x  A    and   x  B  and A – B =   so  A  B   Hence, A B    A – B = .   Let    A B To prove :  A  B = B We can say B   A  B Suppose,...

Q. 16 In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

n(A) = 21                                                                                                       n(B) = 26                                                                             n(C) = 29  n( A  B) = 14 n( A  C) = 12 n (B  C ) = 14 n( A  B  C) = 8       n(liked product C only) = 29 - 4 -8 - 6 = 11 11 people like only product C.

Q3. Let A, B, and C be the sets such that A \cup B = A \cup C and A \cap B = A \cap C. Show that B = C.

       

Let A, B, and C be the sets such that A  B = A  C and A  B = A  C To prove : B = C.  A  B = A + B - A  B =  A  C = A + C - A  C                A + B - A  B =  A + C - A  C                    B - A  B =   C - A  C              ( since  A  B = A  C )                          B = C Hence proved that  B= C.  

Q2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(vi) If A \subset B and x \notin B , then x \notin A

The given statement is true, Let,  A  B and x  B  Suppose, x  A Then, x  B , which is contradiction to x  B Hence, x  A.    

Q.7 Fill in the blanks to make each of the following a true statement :

(i) A \cup A′ = \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot

(ii) \phi ' \capA = \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot

(iii) A \cap A′ =\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot

(iv) U′ \cap A =\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot

The following are the answers for the questions (i) A  A′  U (ii) ′  A   A (iii) A  A′   (iv) U′  A  

Q2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(v) If x \in A and A \not\subset B , then x \in B

The given statement is false, Let        x be 2               A = { 1,2,3}              B = { 4,5,6,7} Here, 2   { 1,2,3} = x  A   and   { 1,2,3}   { 4,5,6,7} = A B But,  2 { 4,5,6,7}  implies  x  B

Q2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(iv) If A \not\subset B and B \not\subset C , then A \not\subset C

The given statement is false Let , A = {1,2}         B = {3,4,5 }         C = { 1,2,6,7,8} Here, {1,2}  {3,4,5 } = A  B  and  {3,4,5 }  { 1,2,6,7,8} = B  C  But ,   {1,2}    { 1,2,6,7,8}  =  A  C

Q. 6 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 60\degree, what is A′?

A' is the set of all triangles whose angle is  in other words A' is set of all equilateral triangles.

Q2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(iii) If A \subset B and B \subset C , then A \subsetC

  Let      A ⊂ B    and     B ⊂ C There be a element x such that     Let,   x  A          x   B            ( Because A  B )         x  C              ( Because B  C ) Hence,the statement is true that   A  C  
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