EDULABZ INTERNATIONAL SETS AND VENN DIAGRAMS

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1 12 SETS ND VENN DIGRMS Section I : Sets 1. Describe the following sets in roster form : (i) 2 { x / x = n, n N, 2 n 5} (ii) {x / x is composite number and 11 < x < 25} (iii) {x / x W, x is divisible by 4 and 6, x 100} (iv) {x / x is two digit number whose sum of digits is 7} n (v) x / x =, n N and n 5 n + 3 2n + 1 (vi) x / x =, n W and n 10 2n + 3 (vii) 2 { x / x W, x < 20} (viii) 2 { x / x = 5 p, P I and x < 400} 1 (ix) x / x =, n N and n 5 n (x) { x /3 x 5 < 15, x W} (xi) {First four planets of our solar system}. ns. (i) {4, 9, 16, 25} (ii) {12, 14, 15, 16, 18, 20, 21, 22, 24} (iii) {0, 12, 24, 36, 48, 60, 72, 84, 96} (iv) {16, 25, 34, 43, 52, 61, 70} (v) ,,,, Math Class VIII 1 Question ank

2 (vi),,,, (vii) { 4, 3, 2, 1, 0, 1, 2, 3, 4} (viii) { 15, 10, 5, 0, 5, 10, 15} (ix) ,,,, (x) {0, 1, 2, 3, 4, 5, 6} (xi) {Mercury, Venus, Earth, Mars} 2. Write the following sets in the builder form : (i) {11, 13, 17, 19, 23, 29, 31, 37} (ii) 1,,,, (iii) {21, 23, 25, 27, 29, 31, 33, 35, 37} (iv),,,, (v) { 10, 5, 0, 5, 10, 15,, 100} (vi) {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}. ns. (i) {x / x is a prime number, 10 < x < 40}. 1 (ii) x / x =, n N and n < 10 n (iii) {x / x = 2n 1, n N, 11 < n < 19} n (iv) x / x =, n is odd natural number n + 2 2n + 1 or x / x =, n W 2n + 3 (v) { x / x = 5 p, p I and 2 p 20} (vi) {x / x N and x is a factor of 48}. Math Class VIII 2 Question ank

3 3. Find, which of the following sets are singleton set : (i) The set of points of intersection of two non-parallel straight lines on the same plane. (ii) = {x : 7x 3 = 11} (iii) = {y : 2y + 1 < 3 and y W} ns.(i) The set of points of intersection of two non-parallel straight lines on the same plane is a singleton set. (ii) = {x : 7x 3 = 11} 7x 3 = 11 7x = x = 14 x = = = {2} Hence, the given set has only one element, so it is a singleton set. (iii) = {y : 2y + 1< 3 and y W } 2y + 1 < 3 2y < 3 1 (Subtracting 1 from both sides) 2y < 2 y < 2 2 (Dividing both sides by 2) y < 1 = {0} Hence, it is a singleton set. 4. State whether the following pairs of sets are equivalent or not : (i) = {x : x N and 11 2x 1} and = {y : y W and 3 y 9} (ii) Set of whole numbers and set of multiples of 3. (iii) P = {5, 6, 7, 8} and M = {x : x W and x 4} ns. (i) = {x : x N and 11 2x 1} 11 2x x ( dding 1 to both sides) Math Class VIII 3 Question ank

4 12 2x 12 2 x 6 x = {1, 2, 3, 4, 5, 6} n() = 6 = {y : y W and 3 y 9} 3 y 9 = {3, 4, 5, 6, 7, 8, 9} n () = 7 Cardinal number of set = 6 and cardinal number of set = 7 Hence, set and set are not equivalent. (ii) Set of whole numbers and set of multiples of 3 are equivalent because both these sets have infinite number of elements. (iii) P = {5, 6, 7, 8} n (P) = 4 M = {x : x W and x 4} M = {0, 1, 2, 3, 4} n(m) = 5 Cardinal number of set P = 4 and Cardinal number of set M = 5. Hence, these sets are not equivalent. 5. () Let = {all quadrilaterals}, = {all rectangles}, C = {all squares} and D = {all rhombuses} in a plane. State, whether the following statements are true or false. (i) C (ii) C (iii) C D (iv) D C (v) C (vi) C ns.(i) False (ii) True (iii) True (iv) False (v) True (vi) False () Let = {all triangles}, = {all isosceles triangles} and C = {all equilateral triangles}. State whether the following statements are true or false. Math Class VIII 4 Question ank

5 (i) C (ii) C ns. (i) False (ii) True 6. Let be the set of letters in the word, seed. Find: (i) (iii) Number of subsets of ns. (i) = {s, e, d} (ii) n () = number of elements = 3 (ii) n () (iv) Number of proper subsets (iii) Number of subsets of = 2 3 = 8 { }, {s}, {e}, {d}, {s, e}, {s, d}, {e, d} {s, e, d}. (iv) Number of proper subsets of = 2 n 1= = 8 1= 7 because { } or φ is not a proper subset. 7. () Find the power set of each of the following sets : (i) = {0, 5} (ii) = {7, 9} (iii) C = {2, 4, 6} ns. (i) P () = {φ, {0}, {5}, {0, 5}}. (ii) P () = {φ, {7}, {9}, {7, 9}}. (iii) P (C) = {φ, {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}}. () Let = {1, {2}}. Find the power set of. ns. (i) P () = {φ, {1}, {{2}}, {1, {2}}. 8. Let 2 2 = { x : x N, x < 50}, = { x : x }, = { x : x = n, n N} and C = {x : x is a factor of 36}. List the elements of each of the sets, and C. lso, state whether each of the following statements is true or false : (i) (ii) = (iii) (iv) C (v) n () < n (C) ns. = { x : x N, x < 50} = {1, 2, 3, 4, 5,, 49, 50} = 2 { x : x } = {1, 2, 3, 4, 5, 6, 7} Math Class VIII 5 Question ank

6 = {x : x = n 2, n N} = {1, 4, 9, 16, 25, 36, 49} C = {x : x is a factor of 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36} (i) False (ii) False (iii) True (iv) False (v) True 9. Let = {letters of OMY} and = {letters of CLCUTT} (i) re these sets disjoint or overlapping? (ii) re these sets equal? (iii) re these sets equivalent? (iv) Is any of these sets, subset of the other? (v) Describe a universal set for this problem. ns. = {letters of OMY} = {,, M, O,Y} = {letters of CLCUTT} = {, C, L, T, U}. (i) These sets are overlapping. (ii) These sets are not equal. (iii) These sets are equivalent. (iv) Neither nor. (v) {Letters of English alphabet}. 10. Let = { x : x W, x < 15}, = {multiples of 2}, = {multiples of 3}, C = {multiples of 5} and D = {multiples of 6} State whether each of the following statements is true or false : (i) C and D are disjoint (ii) C D (iii) C = D (iv) (v) (iv) D (vii) D (viii) C ns. The sets,,, C and D in roster form are = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} = {2, 4, 6, 8, 10, 12, 14} = {3, 6, 9, 12} Math Class VIII 6 Question ank

7 C = {5, 10} D = {6, 12} (i) True (ii) True (iii) False, because C and D are disjoint sets. (iv) False, because has more elements than (v) False, because is not contained in. (vi) True (vii) True (viii) False, because C is not contained in. 11. Given the universal set = { x : x N and x < 20}, find : = {x : x = 3p ; p N} ns. Universal set U = {x : x N and x < 20} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} = {x : x = 3p ; p N} x = 3p If p = 1, then x = 3 1 = 3 If p = 2, then x = 3 2 = 6 If p = 3, then x = 3 3 = 9 If p = 4, then x = 3 4 = 12 If p = 5, then x = 3 5 = 15 If p = 6, then x = 3 6 = 18 = {3, 6, 9, 12, 15, 18} 12. Find the proper subsets of {x : x 2 9x 10 = 0}. ns. x 2 9x 10 = 0 x 2 10x + x 10 = 0 x(x 10) + 1(x 10) = 0 (x 10) (x + 1) = 0 If, (x 10) = 0 x = 10 and, if (x + 1) = 0 x = 1 Given set = { 1, 10} Proper subsets of this set = { 1}, {10}, { 1, 10} 13. Let = {1, 2, 3,, 9}, = {1, 2, 3, 4, 6, 7} and = {4, 6, 8}. Find : (i) (ii) (iii) (iv) ( )' (v) Math Class VIII 7 Question ank

8 ns. = {1, 2, 3, 4, 5, 6, 7, 8, 9} = {1, 2, 3, 4, 6, 7} and = {4, 6, 8} (i) = = {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 6, 7} = {5, 8, 9} (ii) = {1, 2, 3, 4, 5, 6, 7} {4, 6, 8} = {4, 6} (iii) = {4, 6, 8} {1, 2, 3, 4, 5, 6, 7} = { } (iv) ( )' = ( ) = {1, 2, 3, 4, 5, 6, 7, 8, 9} {4, 6} = {1, 2, 3, 5, 7, 8, 9} (v) = {5, 8, 9} {1, 2, 3, 5, 7, 9} = {1, 2, 3, 5, 7, 8, 9} 14. Let = {x : x W, x 10}, = {x : x 5}and = {x : 3 x < 8}. Verify that : (i) ( )' = ' ' (ii) ( )' = ' ' ns.the given sets in the roster form are = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, = {5, 6, 7, 8, 9, 10} = {3, 4, 5, 6, 7} (i) LHS = ( )' = {5, 6, 7, 8, 9,10} {3, 4, 5, 6, 7} = {3, 4, 5, 6, 7, 8, 9, 10} ( ) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {3, 4, 5, 6, 7, 8, 9, 10} = {1, 2} RHS = ' ' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {5, 6, 7, 8, 9, 10} = {1, 2, 3, 4} ' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {3, 4, 5, 6, 7} = {0, 1, 2, 8, 9, 10} ' ' = {1, 2, 3, 4} {0, 1, 2, 8, 9, 10} = {1, 2} Hence, ( )' = ' ' (ii) LHS = ( )' = {5, 6, 7, 8, 9, 10} {3, 4, 5, 6, 7} = {5, 6, 7} Math Class VIII 8 Question ank

9 ( )' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10} {5, 6, 7} = {0, 1, 2, 3, 4, 8, 9, 10} R.H.S. = ' ' ' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {5, 6, 7, 8, 9, 10} = {0, 1, 2, 3, 4} ' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {3, 4, 5, 6, 7} = {0, 1, 2, 8, 9, 10} ' ' = {0, 1, 2, 3, 4} {0, 1, 2, 8, 9, 10} = {0, 1, 2, 3, 4, 8, 9, 10} Hence, ( )' = ' ' 15. Given = {0, 1, 2, 4, 5}, = {0, 2, 4, 6, 8} and C = {0, 3, 6, 9}. Show that : (i) ( C) = ( ) C i. e., the union of sets is associative. (ii) ( C) = ( ) C i.e., the intersection of sets is associative. ns. = {0, 1, 2, 4, 5} and = {0, 2, 4, 6, 8} and C = {0, 3, 6, 9} (i) C = {0, 2, 4, 6, 8} {0, 3, 6, 9} = {0, 2, 3, 4, 6, 8, 9} ( C) = {0, 1, 2, 4, 5} {0, 2, 3, 4, 6, 8, 9} ( C) = {0, 1, 2, 3, 4, 5, 6, 8, 9} (I) = {0, 1, 2, 4, 5} {0, 2, 4, 6, 8} = {0, 1, 2, 4, 5, 6, 8} ( ) C = {0, 1, 2, 4, 5, 6, 8} {0, 3, 6, 9} ( ) C = {0, 1, 2, 3, 4, 5, 6, 8, 9} (II) From (I) and (II), we have ( C) = ( ) C. (ii) C = {0, 2, 4, 6, 8} {0, 3, 6, 9} = {0, 6} Now, ( C) = {0,1, 2, 4, 5} {0, 6} ( C) = {0} (I) Math Class VIII 9 Question ank

10 = {0, 1, 2, 4, 5} {0, 2, 4, 6, 8} = {0, 2, 4} (II) ( ) C = {0, 2, 4} {0, 3, 6, 9} ( ) C = {0} From (I) and (II) we have ( C) = ( ) C 16. If = { x W :5 < x < 10}, = {3, 4, 5, 6, 7} and C = {x = 2n; n N and n 4}. Find : (i) ( C) (ii) ( ) ( C) (iii) ( C) (iv) ( ) ( C) Name the sets which are equal. ns. = {x W :5 < x < 10} = {6, 7, 8, 9} = {3, 4, 5, 6, 7} C = {x = 2n; n N and n 4} x = 2n If n = 1, then x = 2 1 = 2 If n = 2, then x = 2 2 = 4 If n = 3, then x = 2 3 = 6 If n = 4, then x = 2 4 = 8 C = {2, 4, 6, 8} (i) C = {3, 4, 5, 6, 7} {2, 4, 6, 8} = {2, 3, 4, 5, 6, 7, 8} ( C) = {6, 7, 8, 9} {2, 3, 4, 5, 6, 7, 8} ( C) = {6,7,8} (ii) = {3, 4, 5, 6, 7} {6, 7, 8, 9} = {3, 4, 5, 6, 7, 8, 9} ( ) ( C) = {3, 4, 5, 6, 7, 8, 9} {2, 3, 4, 5, 6, 7, 8} = {3, 4, 5, 6, 7, 8} (iii) ( C) = {6, 7, 8, 9} {2, 4, 6, 8} = {6, 8} ( C) = {3, 4, 5, 6, 7} {6, 8} = {3, 4, 5, 6, 7, 8} Math Class VIII 10 Question ank

11 (iv) = {6, 7, 8, 9} {3, 4, 5, 6, 7} = {6, 7} ( ) ( C) = {6, 7} {6, 8} = {6, 7, 8} 17. Let = {x : x N, x < 10}, = {odd numbers}, = {even numbers} and C = {prime numbers}. List the elements of the following sets : (i) ' (ii) ' (iii) C ' (iv) (v) (vi) C (vii) C (viii) (ix) C (x) ( C) ns. = {1, 2, 3, 4, 5, 6, 7, 8, 9} = {1, 3, 5, 7, 9} = {2, 4, 6, 8} C = {2, 3, 5, 7} (i) ' = {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 3, 5, 7, 9} = {2, 4, 6, 8} (ii) ' = {1, 2, 3, 4, 5, 6, 7, 8, 9} {2, 4, 6, 8} = {1, 3, 5, 7, 9} (iii) C ' = {1, 2, 3, 4, 5, 6, 7, 8, 9} {2, 3, 5, 7} = {1, 4, 6, 8, 9} (iv) = {1, 2, 3, 5, 7, 9} {2, 4, 6, 8} = {1, 2, 3, 4, 5, 6, 7, 8, 9} (v) = {1, 3, 5, 7, 9} {2, 4, 6, 8} = φ or { } (vi) C = {1, 3, 5, 7, 9} {2, 3, 5, 7} = {1, 2, 3, 5, 7, 9} (vii) C = {2, 4, 6, 8} {2, 3, 5, 7} = {2, 3, 4, 5, 6, 7, 8} (viii) = {1, 3, 5, 7, 9} {2, 4, 6, 8} = {1, 3, 5, 7, 9} (ix) C= {1, 3, 5, 7, 9} {2, 3, 5, 7} = {1, 9} (x) ( C) = {1, 3, 5, 7, 9} {2, 3, 4, 5, 6, 7, 8} = {3, 5, 7} 18. Let = {x : x W, x 15}, P = {multiples of 2}, Q = {multiples of 3} and R = {multiples of 5}. Show that : (i) ( P Q)' = P Q' (ii) ( P R)' = P' R' (iii) P ( Q R) = ( P Q) ( P R). Math Class VIII 11 Question ank

12 ns. = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} P P = {2, 4, 6, 8, 10, 12, 14} Q = {3, 6, 9, 12, 15} R = {5, 10, 15} (i) L H S = ( P Q)' Q = {2, 4, 6, 8, 10, 12, 14} {3, 6, 9, 12, 15} = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} ( P Q) = ( P Q) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} = {0, 1, 5, 7, 11, 13} R H S = P Q P = {0, 1, 2, 3,..., 15} {2, 4, 6, 8, 10, 12, 14} = {0, 1, 3, 5, 7, 9, 11, 13, 15} Q = {0, 1, 2, 3,..., 15} {3, 6, 9, 12, 15} = {0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14} P Q = {0, 1, 3, 5, 7, 9, 11, 13, 15} {0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14} = {0, 1, 7, 5, 11, 13} Hence, ( P Q) = P Q (Proved) (ii) L H S = ( P R ) P R = {2, 4, 6, 8, 10, 12, 14} { 5, 10, 15} = {10} ( P R) = ( P R) = {0, 1, 2, 3,...,15} {10} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15} R H S = P R P = {0, 1, 2, 3,...,15} {2, 4, 6, 8, 10, 12, 14} = {0, 1, 3, 5, 7, 9, 11, 13, 15} R = R = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} {5, 10, 15} Math Class VIII 12 Question ank

13 = {0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14} P R = {0, 1, 2, 3, 5, 7, 9, 11, 13, 15} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15} Hence, ( P R) = P R (iii) L H S = P ( Q R) = {2, 4, 6, 8, 10, 12, 14} {3, 5, 6, 9, 10, 12, 15} = {6, 10, 12} R H S = ( P Q) ( P R) P Q = {2, 4, 6, 8, 10, 12, 14} {3, 6, 9, 12, 15} = {6, 12} P R = {2, 6, 8, 10, 12, 14} {5, 10, 15} = {10} ( P Q) ( P R) = {6,12} {10} = {6, 10, 12} Hence, P ( Q R) = ( P Q) ( P R). 19. If n () = 20, n () = 16 and n( ) = 30, find n( ). ns.given that, n () = 20, n() = 16 and n( ) = 30 Then n( ) =? We know that n( ) = n( ) + n( ) n( ) 30 = n( ) 30 = 36 n( ) n( ) = n( ) = 6 Hence, n( ) = 6 2 x 20. If = { x : x 16}, = { x : 2 < 3} and the universal set is W, 3 the set of whole numbers. (i) Find sets and. (ii) Verify : = ( ) ns. W = {0, 1, 2, 3, 4, 5, 6, 7, 8, } Math Class VIII 13 Question ank

14 2 = 2 { x : x 16} x 16 x 4 = {0, 1, 2, 3, 4} x = x : 2 < 3 3 x x 2 < 3 < x < 5 x < 15 3 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} (i) = {0, 1, 2, 3, 4} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} (ii) = {5, 6, 7, 8, } = {5, 6, 7, 8,...} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14} (I) = {0, 1, 2, 3, 4} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} = {0, 1, 2, 3, 4} ( ) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} {0, 1, 2, 3, 4} ( ) ={5, 6, 7, 8, 9, 10, 11, 12, 13, 14}... (II) From (I) and (II), we can say that = ( ) 21. If = { x : x 12, x N}; ={x : x is an even number}; = {m : m divisible by 3} and C = {x : 3 < x 9}; then verify that : (i) ( C) = ( C ) (ii) ( C ) = ( C) Math Class VIII 14 Question ank

15 ns. = { x : x 12; x N} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} = {x : x is an even number}= {2, 4, 6, 8, 10, 12} = {m : m is divisible by 3}= {3, 6, 9, 12} C = {x : 3 < x 9} = {4, 5, 6, 7, 8, 9} (i) C = {3, 6, 9, 12} {4, 5, 6, 7, 8, 9} = {3, 4, 5, 6, 7, 8, 9, 12} ( C ) = {1, 2, 10, 11} ( C ) = {2, 4, 6, 8, 10, 12} {1, 2, 10, 11} ( C ) = {4, 6, 8, 12} (I) = {1, 2, 4, 5, 7, 8, 10, 11} C = {1, 2, 3, 10, 11, 12} C = {1, 2, 4, 5, 7, 8, 10, 11} {1, 2, 3, 10, 11, 12} = {1, 2, 10, 11} ( C ) = {2, 4, 6, 8, 10, 12} {1, 2, 10, 11} ( C ) = {4, 6, 8, 12}...(II) From (I) and (II), we have ( C) = ( C ) (ii) C = {1, 2, 4, 5, 7, 8, 10, 11} {1, 2, 3, 10, 11, 12} = {1, 2, 3, 4, 5, 7, 8, 10, 11, 12} ( C ) = {2, 4, 6, 8, 10, 12} {1, 2, 3, 4, 5, 7, 8, 10, 11, 12} ( C ) = {6} (I) Now, C = {3, 6, 9, 12} {4, 5, 6, 7, 8, 9} = {6, 9} ( C ) = {1, 2, 3, 4, 5, 7, 8, 10, 11, 12} ( C ) = {2, 4, 6, 8, 10, 12} {1, 2, 3, 4, 5, 7, 8, 10, 11, 12} Math Class VIII 15 Question ank

16 ( C ) = {6} (II) From (I) and (II), we have ( C ) = ( C) 22. Given = {x : x is a natural number between 25 and 45}; = {x : x is an even number} and = {x : x is a multiple of 3}. Find : (i) n () + n () (ii) n( ) + n( ) (iii) n( ) (iv) n( ) Is n() + n() = n( ) + n( )? Is n( ) = n( )? ns. = {26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44} = {26, 28, 30, 32, 34, 36, 38, 40, 42, 44} = {27, 30, 33, 36, 39, 42} (i) n() = 10, n() = 6 n() + n() = = 16 (ii) We know that n( ) = n( ) + n( ) n( ) n( ) + n( ) = n( ) + n( ) n( ) + n( ) = 16 (iii) = {26, 28, 30, 32, 34, 36, 38, 40, 42, 44} {27, 30, 33, 36, 39, 42} = {26, 28, 32, 34, 38, 40, 44} n( ) = 7 (iv) = {27, 29, 31, 33, 35, 37, 39, 41, 43} = {27, 29, 31, 33, 35, 37, 39, 41, 43} {27, 30, 33, 36, 39, 42} = {27, 33, 39} Math Class VIII 16 Question ank

17 n( ) = 3 Yes, n() + n() = n( ) + n( ) No, n( ) n( ) 23. If n( ) = 30, n() = 22, n() =15 and n( ) = 25; find : (i) n( ) (ii) n( ) (iii) n( ) (iv) n( ) ns. n( ) = 30 n() = 22 n() = 15 n( ) = 25 (i) We know that n( ) + n( ) = n( ) + n( ) 25 + n( ) = n( ) = n( ) = n( ) = 12 (ii) n( ) = n( ) n( ) n( ) = n( ) = 8 (iii) n( ) = n( ) n( ) n( ) = n( ) = 15 (iv) n( ) = 12 n( ) = n( ) n( ) n( ) = n( ) = If n( ) = 40, n( ) = 8 and n ( ) = n ( ), find: (i) n () (ii) n () ns. Given n( ) = 40, n( ) = 8 and n ( )= n ( ) We know that, n ( ) + n ( ) + n ( ) = n ( ) n( ) + n( ) + n( ) = n( ) Math Class VIII 17 Question ank

18 2 n ( ) + 8 = 40 2 n ( )= n ( ) = n( ) = 2 n ( ) = 16 n ( ) = n ( ) = 16 (i) n( ) = n( ) + n( ) = = 24 (ii) n( ) = n( ) + n( ) = = If n( ) = 40, n( ) = 15, n( ) = 12 and n(( ) ) = 32, find : (i) n() (ii) n( ) (iii) n( ) (iv) n( ) (v) n( ) (vi) n( ) ns. n( ) = 40, n( ) = 15, n () = 12 and n(( ) ) = 32 (i) n( ) = n( ) n( ) = = 25 (ii) n( ) = n( ) n( ) = = 28 (iii) n( ) = n( ) n(( ) ) = = 8 (iv) n( ) = n( ) + n( ) n( ) = = = 29 (v) n( ) = n( ) n( ) = 25 8 = 17 (vi) n( ) = n( ) n( ) = 12 8 = If n ( ) = 12, n ( ) = 16 and n( ) = 5, find : (i) n () (ii) n () (iii) n( ) ns. n ( ) = 12, n ( ) = 16 and n( ) = 5 (i) n( ) = n( ) + n( ) [ n( ) = n( ) n( )] = = 17 (ii) n( ) = n( ) + n( ) [ n( ) = n( ) ( )] = = 21 (iii) n( ) = n( ) + n( ) n( ) = = 38 5 = 33 Math Class VIII 18 Question ank

19 Section II : Venn diagrams 1. From the adjoining Venn diagram, find the following sets: (i) (ii) (iii) C (iv) C (v) C (vi) C (vii) C (viii) ( ) (ix) ( C ) ns. (i) = {0,1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12} (ii) = {8, 0, 5} (iii) C = {0, 5} (iv) C = {2, 7, 8, 9, 10, 11, 12} (v) C = {8, 10} (vi) C = {7, 8, 11} (vii) C = {3, 4, 6} (viii) ( ) = {2, 4, 6, 9, 12} (ix) ( C ) = {2, 9, 12} 2. From the given diagram find: (i) (ii) (iii) (iv) (v) ( ) ns. (i) = {a, c, d, e} {b, c, e, f} = {a, b, c, d, e, f} (ii) = {b, f, g, h} = {b, f, g, h} {b, c, e, f} = {b, f} (iii) = {a, c, d, e} {b, c, e, f} = {a, d} (iv) = {b, c, e, f} {a, c, d, e} = {b, f} (v) = {a, b, c, d, e, f} ( ) = {h, g} 2 h a d 4 6 C c e b f g 9 Math Class VIII 19 Question ank

20 3. From the given diagram, find: (i) (ii) (iii) (iv) ( ) Is = ( )? lso, verify if = ( ) ns. (i) = {1, 3, 4, 6} = {2, 5, 7, 8, 9, 10} (ii) = {1, 2, 5} = {3, 4, 6, 7, 8, 9, 10} (iii) = {2, 5, 7, 8, 9, 10} {3, 4, 6, 7, 8, 9, 10} = {2, 3, 4, 5, 6, 7, 8, 9, 10} (iv) = {1, 3, 4, 6} {1, 2, 5} = {1} ( ) = {2, 3, 4, 5, 6, 7, 8, 9, 10} From Part (iii) and Part (iv) we conclude = ( ) Now = {2, 5, 7, 8, 9, 10} {3, 4, 6, 7, 8, 9, 10} = {7, 8, 9, 10}... (I) Now = {1, 3, 4, 6} {1, 2, 5} = {1, 2, 3, 4, 5, 6} ( ) = {7, 8, 9, 10}...(II) From (I) and (II), we have = ( ) a d e 4. Use the given diagram to find: g b c h f (i) ( C) p i j (ii) ( C) k l n C (iii) (iv) m Is =? ns. (i) C = {d, e, f, g, h, j} {h, i, j, k, l} = {h, j} ( C) = {a, b, c, d, g, h, i} {h, j} = {a, b, c, d, g, h, i, j} (ii) C = {a, b, c, d, g, h, i} {h, i, j, k, l} = {a, b, c, d, g} Math Class VIII 20 Question ank

21 ( C) = {d, e, f, g, h, j} {a, b, c, d, g} = {e, f, h, j} (iii) = {a, b, c, d, g, h, i} {d, e, f, g, h, j} = {a, b, c, i}...(i) (iv) = {a, b, c, i, k, l, m, n, p} = {a, b, c, d, g, h, i} {a, b, c, i, k, l, m, n, p} = {a, b, c, i} (II) From (I) and (II) we can conclude = 5. Draw a Venn-diagram to show the relationship between two overlapping sets and. Now shade the region representing : (i) (ii) ns.(i) = (ii) = (iii) = (iii) 6. Draw a Venn-diagram to show the relationship between sets and ; such that. Now shade the region representing : (i) (ii) (iii) (iv) ( ) ns.(i) = Math Class VIII 21 Question ank

22 (ii) = (iii) = (iv) ( ) = 7. Two sets and are such that = φ. Draw a Venn-diagram to show the relationship between and. Shade the region representing : (i) (ii) ( ) (iii) ns. (i) = (ii) ( ) = (iii) = 8. () State the sets represented by the shaded portion of following Venn-diagrams : Math Class VIII 22 Question ank

23 (i) (ii) (iii) ns. (i) ( ) (ii) or (iii) ( ) () In each of the given diagrams, shade the region which represents the set given underneath the diagram : (i) (ii) (iii) P ( ) (P Q) ns. (i) ( ) = Q ( ) (ii) ( ) = Math Class VIII 23 Question ank

24 (iii) ( P Q ) = P Q 9. In the adjoining figure, and are two subsets of the universal set such that, n( ) = 41, n( ) = 25 and n( ) = 50. Find: (i) n( ) (ii) n( ) (iii) n( ) ns. (i) n( ) = n( ) n( ) = = 9 (ii) n( ) = n( ) n( ) = = 25 (iii) n( ) = n( ) n( ) = n( ) n( ) [ n( = n()] = = If = { x : x N and x 20}, = {x : x is a multiple of 4}, = {x : x is a multiple of 6} and C = {x : x is a factor of 36}. Draw a Venndiagram to show that the relationship between the given sets. Hence, find : (i) C (ii) (iii) C ns. = { x : x N and x 20} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, , 14, 15, 16, 17, 18, 19, 20} = {x : x is a multiple of 4} C 15 = {4, 8, 12, 16, 20} = {x : x is a multiple of 6} = {6, 12, 18} C = {x : x is a factor 36} = {1, 2, 3, 4, 6, 9, 12, 18} (i) C = {4,12} (ii) = {4, 8, 16, 20} (iii) C = {12} 11. In a class of 60 pupils, 28 play hockey, 33 play cricket and 14 play none of these games. Draw Venn diagram to find : (i) Pupils play both the games. (ii) How many play hockey only? (iii) How many play cricket only? Math Class VIII 24 Question ank

25 ns.let H for hockey and C for cricket n(h) = 25, n(c) = 33 and 14 play none of these games i.e., n( H C) = = 46 (i) n( H C) = n( H) + n( C) n( H C) = = = 15 (ii) n( H) n( H C) = = 13 (iii) n( C) n( H C) = = In a club, three-tenths of its members play cards only, four-tenths play carrom only. If 15 members play none of these games and 90 play both, find using Venn diagram, the total number of members in the club. ns. Let number of members in the club be x. Number of members play card only Number of members play carrom only = 3 10 x = 4 10 x 3 x 4 x = x X Y 3x 4x 3x 4x x = x 3 x 4x = x = x = x = Hence, total number of members in the club are In a colony, two-fifths of the families read the magazine Femina and three-fourths of the families read the magazine Filmfare. If 40 families read none of these two magazines and 100 families read both, use Venn-diagram to find the number of families in the colony. Math Class VIII 25 Question ank

26 ns. Let number of families in the colony be x. 2 Families reading Femina = 5 x X Y 3 Families reading Filmfare = 4 x 2x x The Venn diagram is as given alongside 2 x x = x x x = x 5 4 2x 3 x + 60 = x 2 x 3 x + x = x + 15 x 20x = x 20 x = x = 60 x 60 = 20 = Hence, total number of families in the colony is In a class of 50 boys, 35 like horror movies, 30 like war movies and 5 like neither. Find the number of those that like both. ns.let H denote for horror movies and W denote for war movies n (H) = 35, n(w) = 30 and 5 like neither i.e., n( ) = 50 5 = 45. Hence, number of boys like both n( H W ) n( H) + n( W ) n( H W ) = = = In a certain locality of Delhi there are 1000 families. survey showed that 504 subscribe to The Hindustan Times daily newspaper and 478 subscribe to The Times of India and 106 subscribe to both. Find the number of families which do not subscribe to any of these newspapers. ns.total number of families are 1000 Let be the set of families who subscribe The Hindustan Time, is the set of families who subscribe The Times of India. Math Class VIII 26 Question ank

27 Then, n() = 504, n() = 478 and n( ) =106 Those families who subscribe to either the Hindustan Times or to the Times of India or both = n( ) = n( ) + n( ) n( ) = = = 876 Hence, those families who subscribe to none = In a class of 90 students, 50 students got distinction in Mathematics, 42 got distinction in Science and 24 students got distinction in both the subjects. Represent this information by a Venn diagram. Hence, find the number of students getting distinction in (i) Mathematics only (ii) Science only (iii) any of the two subjects (iv) neither Mathematics nor Science ns.let U be all the students in class. X be the students who gots distinction in Mathematics Y be the students who gots distinction in Science n (U) = 90 U X Y 22 n(x) = n (Y) = 42 n( X Y ) = 24 Thus, number of students getting distinction in (i) Maths only = 26 (ii) Science only = 18 (iii) any of the two subjects = = 68 (iv) neither Mathematics nor Science = In a group of 60 persons, 45 speak engali, 28 speak English and all the persons speak at least one language. Find how many people speak both engali and English. Draw a Venn diagram. ns. Let U = group of persons Math Class VIII 27 Question ank

28 X = Those persons who speak engali Y = Those persons who speak English n (U) = 60 n (X) = 45 n (Y) = 28 and n( X Y ) = 60 n( X Y ) = n( X ) + n( Y ) n( X Y ) = = Forty-three persons went to a canteen, which sells soup and tea. If 18 persons took soup only; 8 took tea only and 5 took nothing. Use Venn diagram to find : U X Y (i) how many took both? (ii) how many took soup? (iii) how many took tea? 5 ns. Total number of persons who visited canteen = 43 Let U represents the set of persons who visited canteen. Number of persons who took soup only = 18 Number of persons who took tea only = 8 Number of persons who took nothing = 5 Number of persons who took either of two = 43 5 = 38 Number of persons who took both = 38 (18 + 8) = = 12 (i) Number of those who took both = 12 (ii) Number of those who took soup = = 30 (iii) Number of those who took tea = = If is the set of boys in your school and is the set of boys who play badminton. Draw Venn diagram showing that some of boys do not play badminton. If n( ) = 40 and n( ) = 17; find : (i) how many do not play badminton? (ii) how many play badminton? U X Y Math Class VIII 28 Question ank

29 ns. = Set of boys in the school = Set of boys who play badminton = Set of boys who do not play badminton = Shaded Portion Now, if n( ) = 40 n( ) = 17 Then (i) number of oys who do not play badminton = 17 (ii) Number of boys who play badminton = = Let = {all triangles drawn in plane}, I = {isosceles triangles} and R = {right angled triangles} Draw a Venn diagram to show these sets in their correct relationship. Shade the region representing I R and write the measures of the angles of the triangles of this region. ns. The given sets are I R = {all triangles drawn in a plane} I = {isosceles triangles} R = {right angled triangles} I R Measures of angles of the triangles in the region I R = 45, 45, In a city, 50% people read newspaper, 45% read newspaper, and 25% read neither nor. What percentage of people read both the newspapers as well as? ns. Let = Those people who read newspaper = Those people who read newspaper Here, n( ) = 100%, n( ) = 50%, n( ) = 45%, and n( ) = 25% n( ) = n( ) n(( ) ) =100% 25% = 75% We know that n( ) = n( ) + n( ) n( ) 75% = 50% + 45% n( ) Math Class VIII 29 Question ank

30 n( ) = 95% 75% n( ) = 20% 22. Preeti and Rashmi contested the selection for the post of head girl of the school, for which the students of classes 10 th, 11 th and 12 th voted. If three-seventh of the students voted for preeti only; three-seventh for Rashmi; fifty for both and 50 students did not use their votes, find : (i) the total number of students in classes 10 th, 11 th and 12 th, (ii) the number of students, who voted for Preeti and (iii) the number of students, who voted for Rashmi only. ns. Let total number of students in classes 10 th, 11 th, and 12 th be x Let represents the set of students in classes 10 th, 11 th and 12 th 3x Number of students who voted for Preeti only = 7 Number of students who voted for both (Preeti and Rashmi) = 50 3x Number of students who voted for Rashmi = 7 3 x Number of students who voted for Rashmi only = 50 7 Number of those who did not use their voted = 50 From Venn diagram, 3x 3x x = x 3x x = x 3 x 3x = x x = 50 7 x = 7 50 x = 50 7 = 350 X 3x x 7 Y Math Class VIII 30 Question ank

31 (i) Total number of students in classes 10 th, 11 th and 12 th = 350 (ii) Number of students who voted for Preeti 3x = + 50 = = = = 200 (iii) Number of students who voted for Rashmi only 3 x = 50 = = = The students of a certain school have a choice of three games : Tennis, adminton and Cricket. The following table gives the percentage of students who play some or all the games : Games Tennis adminton Tennis adminton Cricket Cricket ll and and and only Games adminton Cricket Tennis % of students Draw a Venn diagram and use it to determine the percentage of students who (i) play Tennis only (ii) play adminton only (iii) play Cricket (iv) do not play any of the games. ns. Let X = Those students who play Tennis Y = Those students who play adminton Z = Those students who play Cricket. u Y Then from given data 20 7 X n (U) = 100% n (X) = 35% 30 n (Y) = 30% Z n( X Y) = 10% n( Y Z) = 10% n( X Z) = 8% Math Class VIII 31 Question ank

32 n (only Z) = 30% n( X Y Z) = 3% Using Venn diagram (i) Play Tennis only = 20% (ii) Play adminton only = 13% (iii) Play Cricket = 45% (iv) Do not play any of the games = 15% 24. and are two overlapping sets such that n( ) = x + 4, n ( ) = 4x 8 and n () = 3x + 8. Find x, if n( ) = 70. lso find n( ). ns. n( ) = 70 n( ) = x + 4 n( ) = 4x 8 n () = 3x + 8 n ( ) = n () n( ) = 3x + 8 (x + 4) = 3x + 8 x 4 = 2x + 4 Using Venn diagram, we have 4x 8 + x x + 4 = 70 4x + 2x + x = x = 70 x = 70 7 = 10 n( ) = n (only ) = n ( ) = 2x + 4 = = = 24 4x 8 x + 4 2x + 4 Math Class VIII 32 Question ank