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9MathQ01-3,5
Chapter 1
Sets and Functions
Exercise 1.1

1. Write each of the following sets (i) in set builder form (ii) in listing its elements.
   (1) The set N of natural numbers.
   (2) The set J of all positive integers.
   (3) The set P of all prime numbers.
   (4) The set A of all positive integers that lie between 1 and 13.
   (5) The set B of real numbers which satisfy the equation 3x2 + 5x – 2 = 0.
2. Choose a suitable description (a) of (b) or (c) in set builder form for the following sets.
   (1) E ={ 2, 4, 6, 8}
   (a) E = { x/ x is an even integer less than 10 }
   (b) E = { x/ x is an even positive integer less than 10 }
   (c) E = { x / x is positive integer, x< 10 and x is a multiple of 2}
   (2) F = { 3, 6,9, 12, 15 , …}
   (a) F = {x/ x is appositive integer that is divisible by 3}
   (b) F = {x/x is a multiple of 3}
   (c) F = {x/x is a natural number that is divisible by 3}
3. A = {x/x2 + x – 6 } and B = { -3,2}. Is A = B?
4. A = {x/x is prime number which is less than 10} and B = {x/x2 – 8x + 15 = 0}
   (a) Is A = B (b) Is B⊂ A?
5. P = {x/x is an integer and -1 < x<3/5 }and Q = {x/x3 -3x2 + 2x = 0} . Is P = Q?
   6.L= {(x,y)/ x and y are positive integers and x + y = 7}.Write L by listing its elements.

Exercise 1.2
1.Draw the following intervals.
(a) {x/x > 2} (b) {x/x ≥ 3} (c) {x/x x ≤ -1} (d) {x/x>-1}
(e){x/-2≤ x≤ 2} (f) {x/0≤x≤ 5} (g) {x/x≤0 or x.2}

1. Draw a graph to show the solution set of each of the following.
   (a) x-1<4 (b) x-1≤ 0 (c) 2x≤5 (d) 2x-1>7
   (e) 5-x≥1 (f) 1/3(x-1)<1
   3.Draw the graph of the following number lines below one another.
   (a) P = {x/x≥3, x∈R} (b) Q = {x/x≤-2, x∈R}
   (c) P∩Q (d) P∪Q
2. On separate number lines draw the graph.
   S = {x/x>-4} , T = {x/x<3}.Give a set –builder description of S∩T.

Exercise 1.3

1. M = {x/x is an integer , and -3<x<6} , N = the set of positive integers that are less than 8.
   Find M∩N. (3 marks)
2. A = {x/x is a positive integer that is divisible by 3}, B = {x/x is a positive integer that is
   divisible by 5. Find (a) A∩B (b) L.C.M of 3 and 5
3. J = {1,2,3,4,……} the set of positive integers and P = {x/x is a prime number} ,find J∩P.
   4.A = {x/x is a positive even integer }. B = { x/x is a prime number}. C = { x/x is a positive
   integer that is divisible by 3}. Find (a) A∩ (B∩C) and (A∩B) ∩C.
   Show that A∩ (B∩C)= (A ∩ B) ∩ C

1. Let A = {x/x is positive integer that is divisible by 2}. B = { x/x is a positive integer that is
   divisible by 3}. C = {x/x is appositive integer that is divisible by 5}. List the elements of the
   sets A, B, C. Find (a) A∩ (B∩C) and (b) (A∩B) ∩C. Show that A∩ (B∩C)= (A ∩ B) ∩ C
   6.Let A = {x/x is a positive integer less than 7} and B = {x/x is an integer land -3 ≤ x ≤4}.
   List the number of A and B, and then write down A ∪ B.
2. Let A = {x/x is an integer, 0<x<6} and B = {x/x is a positive integer less than 13 and x is
   a multiple of 3}. List the number of A and B, and find A ∪ B.
3. If P = {x/x2 + 2x -3} and Q = {x/x is an integer, -1 ≤ x ≤ 4}, find P ∩ Q and P ∪ Q.

Exercise 1.4

1. Find the set B\A and A\B in the following.
   (a) A = { 1,2,3,4,5,6} , B = { 3,5} (b) A = { p,q,r,s} , B = { x,y,z}
   (c) A = { 1,2,3} , B = { 1,2,3,4}
2. S = the set of natural numbers and T = the set of positive even numbers. Describe in words
   the complement of T.
3. Let S = {1,2,3,4,5,6,7,8,9} and A = { 1,2,3,4,5}, B = { 2,4,6,8}. Find (a) A^' (b) B^'
   (c) A^'∩B^' (d) A∪B (e) (A∪B)^' what can you say about A^'∩B^' and (A∪B)^'?
4. S = {1,2,3,4,5,6,7,8}, A = {1,2,3,5,7}, B = {1,3,5,7}, C = {2,4,6,8}
   (i) List the sets A^', B^'and C^'.
   (ii) State whether each of the following is true or false.
   (a) A∩A^'=∅ (b) A∪A^'=∅ (c) A⊂B (d) B⊂A
   (e) A^'⊂B^' (f) B^'⊂A^' (g) B^'=C (h) B⊂C^'

Exercise 1.5
1.A and B are two sets and numbers of elements ate as shown in the Venn diagram. Given that
n(A) = n(B), calculate (a) x (b) n(A∪B)

2.A ,B and C are three sets and the number of elements are shown in the Venn diagram. Given
that the universal set S = A∪B∪C and n(S) = 34, find (a) the value of x (b) n(A∩B∩C^' ).

1. S= {x\x is an integer, 1≤x≤10}, A = {x\x – 1 ≥3}, B = {x\8<4x<30}.
   List the elements of sets S,A,B.
   Verify (a) (A∪B)´ = A´∩B´ (b) (A∩B)´ =A´∩ B´

2. S = {2,3,4,5,6,7,8,9,10,11}, A = {x\x is a factor of 18}, B = {x\3x – 1 > 20}.
   Verify that (a) De Morgan’s Laws (b) A\B = A∩B´

3. Given that S = { x\x is an integer, 30≤x≤50}. A k= { x\ x is a prime number , x>30}
   b = { x\x is a multiple of 5}, C = { x\x is an odd number}
   (a) Find A∩B,A∪C
   (b) Verify that n(B∪C) = n(B) + n(C) – n(B∩C)
   (c) A´\C = A´∩C´

4. Let S = {1,2,3,4,5,7}, A = { x\x is a prime number}, B = { x\ 1<x<5}.Find A×B, B×A.

5. If A = {1,2}, B = {2,3,4}, find (B\A)×(A∪B) (3 marks)

Exercise 1.6

1. S = {0,1,2,5} and T = { 2,3,4,6,7}. The given relation from S to T “is one less than”.
   Draw the corresponding arrow diagram. (3 marks)
2. Draw the arrow diagram to describe the relation “is a factor of” from set A ={2,3,5,7,11}
   to the set B = {1,6,12,17,30,35}.
3. Draw ten different arrow diagrams for the sets A and B shown in the following figure.

Exercise 1.7

1. A = {2,3,5,6} and B = { 1,2,3,4,5,6}. Take the relation “is a factor of” from A to B. Draw an
   arrow diagram . Write down the set of ordered pairs which describe this relation.
2. X = { 2,6,8} and Y = {1,3,5}. Let the given relation from X to Y be “x is two times y” where
   x∈X and y∈Y. Write down the set of ordered opairs which describes this relation. Draw the
   graph of this relation.
   3.A = B = {1,2,3,6}. Take the relation “is a factor of” from A to B. Draw an arrow diagram.
3. Write down the set of ordered pairs which describes the relation “is less than” from A to B,
   where A = B = {,3,4}. A relation from X to Y is given by the set of ordered pairs
   {(a,3),(b,2),(b,4),(c,1)}. Draw the arrow diagram. Is it a function? Why?
4. S = {1,2,3,4} and T = {8,9}. If the image is of each odd number in S is 9 and the image of
   each even number is 8, describe the function by an arrow diagram. Write down the set of
   ordered pairs for this function.
5. Let S = {1,2,3,……,20}. A relation from S to S is described by the set of ordered pairs (x,y)
   where y∈S, x∈S and y = 3x. Write down all the ordered pairs for this relation. If this function ?
   Why?
6. K = {1,2,3,4,5,6}. A function from K to K is such that for n∈K
   n ↦1 if n is odd , and
   n↦ n/2 if n is even
   Describe this function by the set of ordered pairs.
7. E = {3,4,5,…..21} and F = {2,4,6,8,10}.
   A function F to E is such that n↦ n+3 for each n∈F. Find the set of ordered pairs for the
   function. Write down the range.
8. Describe all the possible functions from A = {a,b,c} to B = {p,q} by drawing arrow diagram.

Exercise 1.10
1.Make a table for the function x⟼x + 1 for the set { 0,1,2,3,4} to the set Z. Draw the graph.

1. Make tables for the functions x⟼2x, x⟼3x, x⟼4x from the set {0,1,4} to the set Z. Draw
   the graphs of these functions .
2. Make table for the function x⟼x from the set {0,1,2,3} to the set Z, and draw a graph.
   Draw a smooth curve through the points.
3. Make tables for the functions.
   (a) x⟼ x, (b) x ⟼ x, (c) x ⟼ x + 2, (d) x ⟼ -x+2 (e) x ⟼ 2x-1
   from the set A = {-5,-4,-3,-2,-1,0,1,2,3,4,5} to the set Z. Draw a graphs.
4. Draw the graph of the relation
   R = {(-4,-3),(-3,--2),(-2,-1),(-1,0),(0,1),(1,2),(2,3),(3,4)}.

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