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Sets and Subsets| Set Operations| Functions| Cartesian Product| Relations 

 

SETS AND SUBSETS

Exercise 1. Let A={a, b, {c, d}, e}. How many elements does A contain?

 

Exercise 2. Let A = {2, {4, 5}, 4}. Which statement is correct?

a) 5 is an element of A.

b) {5} is an element of A.

c) {4, 5} is an element of A.

c) {5} is a subset of A.

 

Exercise 3. Which of these sets is finite?

a) {x | x is even}

b) {x | x < 5}

c) {1, 2, 3,...}

d) {1, 2, 3,...,999,1000}

 

Exercise 4. Which of these sets is not a null set?

a) A = {x | 6x = 24 and 3x = 1}

b) B = {x | x + 10 = 10}

c) C = {x | x is a man older than 200 years}

d) D = {x | x < x}

 

Exercise 5. Let S={1, 2, 3}. How many subsets does S contain?

 

Exercise 6. Let D E. Suppose a D and b E. Which of the following statements must be true?

a) c D

b) b D

c) a E

d) a D

 

Exercise 7. Let A = {x | x is even}, B = {1, 2, 3,..., 99, 100}, C = {3, 5, 7, 9}, D = {101, 102} and E = {101, 103, 105}. Which of these sets can equal S if S A and S and B are disjoint?

a) A     b) B     c) C     d) D     e) E

Exercise 8. Which statement best describes the Venn diagram below?

a) A = B

b) A and B are not comparable

c) A B

d) A B

 

Exercise 9. Let S = {a, b}. How many elements does the power set 2S contain?

 

Exercise 10. Which set S does the power set 2S = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} come from?

a) {{1},{2},{3}}

b) {1, 2, 3}

c) {{1, 2}, {2, 3}, {1, 3}}

d) {{1, 2, 3}}

 

SOLUTIONS

Solution 1

A contains 4 elements: a, b, {c, d}, e

Solution 2

A is consisted of elements: 3, {4, 5} and 8, so {4,5} is an element of A

Solution 3

{1,2,3, ..., 1000} is finite, because it is consisted of final number of elements.

Solution 4

Set B is not an empty set because it contains one element. The only element of the set B is zero.
B = {0}

Solution 5

Set {1, 2, 3} contains 8 subsets: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1,2,3}, {}.

Solution 6

aE is true, because  aD and  DE means that every element from D is contained in E.

Solution 7

The correct answer is E, because E consists of even numbers as elements and the intersection of sets S and B is a null set.

Solution 8

A  B is the the correct answer because A is a superset of B.

Solution 9

The power set 2S contains 4 elements: {a, b}, {a}, {b}, {}.

Solution 10

The correct answer is {1, 2, 3} because all subsets of {1, 2, 3} are  , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}. 

 

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SET OPERATIONS

Exercise 1. Let A = {x, y, z}, B = {v, w, x}. Which of the following statements is correct?

a) A  B = {v, w, x, y, z}

b) A  B = {v, w, y, z}

c) A  B = {v, w, x, y}

d) A  B = {x, w, x, y, z}

 

Exercise 2. Let A = {1, 2, 3, ..., 8, 9} and B = {3, 5, 7, 9}. Which of the following statements is correct?.

a) A B = {2, 4, 6}

b) A B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

c) A B = {1, 2, 4, 6, 8}

d) A B = {2, 4, 6, 8}

 

Exercise 3. Let C = {1, 2, 3, 4} and D = {1, 3, 5, 7, 9}. How many elements does the set C D contain?

How many elements does the set C D contain?

 

Exercise 4. Let A = {2, 3, 4}, B = {3} and C = {x | x is even}. Which statement is correct?

a) C A = B

b) C B = A

c) A C

d) C / A = B

 

Exercise 5. Let A B, B C and D A = C. Which statement is always false?

a) B D

b) A C

c) A = B

b) B D =  and B A

 

Exercise 6. What is shaded in the Venn diagram below?.

 

a) A B

b) A B

c) A

d) B

 

Exercise 7. What is shaded in the Venn diagram below?.

a) A B

b) A'

c) A - B

d) B - A

 

Exercise 8. Let U = {1, 2, 3, ..., 8, 9} and A = {1, 3, 5, 7}. Find A'.

a) A' = {2, 4, 6, 8}

b) A' = {2, 4, 6, 8, 9}

c) A' = {2, 4, 6}

d) A' = {9}

 

Exercise 9. Let U = {1, 2, 3,..., 8, 9}, B = {1, 3, 5, 7} and C = {2, 3, 4, 5, 6}. How many elements does the set (B C)' contain?

How many elements does the set (C - B)' contain?

 

SOLUTIONS

Solution 1

The union of sets A and B is the set of all elements which belong to A or to B or to both.  So, AB = { v, w, x, y, z } because it contains all elements from A and all elements from B.

Solution 2

AB = { 3, 5, 7, 9 } is the correct answer because the intersection of sets A and B is the set of elements which are common to A and B.

Solution 3

CD = { 1, 2, 3, 4, 5, 7, 9 } contains 7 elements. CD = { 1, 3} contains 2 elements.

Solution 4

CA = B is the correct answer. C has only even elements and three is the only even element in A. As three belongs to B and is the only element of B, the intersection of C and A must be the set B.

Solution 5

BD = and BA  is correct. As A is a proper subset of B, there are elements from B in C that are not in A, so BA . As DA = C, then must be BD = .

Solution 6

The  shaded area is the common area of A and B. The intersection of sets A and B is the set of elements which are common to A and B.

Solution 7

B - A is the correct answer. The shaded area is the difference of sets B and A, which is the set of elements which belong to B, but which do not belong to A.

Solution 8

Ac = {2, 4, 6, 8, 9} is the correct answer. Ac   is set of those elements in U which are not in A. We only have to list all the elements in U that are not in A.

Solution 9

BC = { 3, 5 }; (BC)c = { 1, 2, 4, 6, 7, 8, 9 } and that counts 7 elements.

C - B = { 2, 4, 6 }; (C - B)c =  { 1, 3, 5, 7, 8, 9 } and that counts 6 elements.

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FUNCTIONS

Exercise 1. Which of the diagrams defines a function of A = {a, b, c, d} into B = {1, 2, 3}

a)b)c)d)

 

 

Exercise 2. Let f(x) = x2 . Find the value of f (5).

f(5) =

 

Exercise 3. Let R be the set of real numbers and let the function f : R  R be defined by

Find the values of f(5) and f (6).

f(5) =

f(6) =

 

Exercise 4. Let A = {Mary, Jane, Ann} and B = {Pizza, Lasagne}. How many different functions are there from A into B?

 

Exercise 5. The following diagram defines the function f, which maps the set {a, b, c, d, e} into itself. How many elements does the range of f contain?

 

Exercise 6. Let f 1, f 2, f 3, f 4, f 5 be functions of R into R and let f 1(x) = x2 + 3x - 4. Which of these functions are equal to f 1?

a) f2(x) = x2

b) f3(y) = y2 - 4

c) f4(z) = z2 + 3z - 4

d) f5(x) = x2 + 3x

 

Exercise 7. Let the functions f, g, and h be defined by
(a) f(x) = x3where x R
(b) g (y) = y3 where 2 < y < 9
(c) h(z) = z 3 where 1 < z < 10
Which of these functions are equal?

 

Exercise 8. Which of the following functions is one-one?

a) To each show assign its first performance.

b) To each student assign his mentor.

c) To each pair of shoes assign its price.

d) To each school assign the number of computers it has.

 

Exercise 9. Let A = {1, 2, 3, 4}. Let f, g and h be functions of A into R. Which one of them is one-one?

a) f(1) = 3, f(2) = 4, f(3) = 5, f(4) = 3

b) g(1) = 2, g(2) = 4, g(3) = 5, g(4) = 3

c) h(1) = 2, h(2) =4, h(3) = 3, h(4) = 2

 

Exercise 10. Let A = [-2,-3], B = [-3,5] and C = [2,-2]. Which of the following functions is not one-one?

a) f : A R

b) g : B R

c) h : C R

 

Exercise 11. Let A = [-1, 1]. which of these functions is onto?

a) f(x) = x2

b) g (x) = x3

c) h(x) = x4

 

Exercise 12. Let the functions f : R R and g : R R be defined by f(x) = 2x g(x) = 3

Find the composition function f g .

f    g =

 

Exercise 13. Let the functions f  : R R be defined by f (x) = x3. Find f -1 (27)

 

Exercise 14. Let A = [-1, 1]. Let functions f , g and h< > be functions of A into A. Which function has an inverse function?

a) f (x) = sin x

b) g(x) = x4

c) h(x) = x3s

 

SOLUTIONS

Solution 1. B is the correct answer because every element from A has its image.

Solution 2. (5)=25 The image of 5 is 25 i.e. f  : 5 25.

Solution 3. For x > 5, f (6) = 12-3, f (6)= 9 For x 5, f (5)= 20+5, f (5)=25< >

Solution 4. Each function assigns either pizza or lasagne, but not both, to each element in the set A, so there are 8 functions from A into B.  

Solution 5. b,d,e are images, so the range of f  is the set {b, d, e} that consists of 3 elements.

Solution 6. Correct answer is f 4(z) = z2 + 3z - 4 , both functions are defined on the set of real numbers. 

Solution 7. None of the functions are equal, because they have different domains. 

Solution 8. There are no two different shows that have the same first performance, so this function is injection.

Solution 9 g is injection. f is not injection because f (1) = f (4) and h is not injection because h(1)=h (4).   

Solution 10. h is injection.

Solution 11. Function g is bijection. f (-1) = f (1) so f  is not bijection and h is not bijection since h(-1) = h (1).

Solution 12. (f g)(x) = f (g(x)) = f  (3) = 2  3 = 6

Solution 13. f -1(27)= {3} since f (3)= 27

Solution 14. f (x)=sin x is not bijection so f  has no inverse. g(x)=x4 has no inverse since g is neither one-one or onto. h(x)=x3 has an inverse function because  f is injection since x y x3 y3. 

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CARTESIAN PRODUCT

Exercise 1. Find the ordered pairs corresponding to the points A and B, which appear in the coordinate diagram {1, 2, 3} × {1, 2, 3} below.


a) A = (2, 1), B = (1, 3)

b) A = (1, 2), B = (3, 1)

c) A = (3, 1), B = (1, 2)

d) A = (2, 2), B = (2, 3)

 

Exercise 2. Find the values of x and y if the ordered pairs (x + y, 1) and (7, x - y) are equal.

x =

y =

Exercise 3. Suppose sets A, B and C have 2, 3 and 4 elements respectively. How many elements are there in A × B × C ?

Exercise 4. Let D = {1, 2}, E = {2, 3} and F = {4, 5}. How many elements are there in (D E) × F? 

Exercise 5. Let R be the set of real numbers and let the function f : R R be defined by f (x) = x - 2. Which ordered pair belongs to the graph f * of the function f ?

a) (5, 6)

b) (3, 2)

c) (5, 3)

d) (4, 1)

 

Exercise 6. Let S = {a, b, c, d}. Which of the following sets of ordered pairs is a function of S into S?

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

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

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

d) {(d, b), (c, a), (b, e), a, c)}

Exercise 7. Let A = {1, 2, 3}. Which of the following diagrams of A × A is a function from A into A? (a) (b) (c)

Exercise 8. Let B = 1, 2, 3 and let f  and g be functions of A into A . Let f  be the set of points displayed in th first diagram and g be the set of points displayed in the second diagram. Find the value of f(g(1)).

Exercise 9. From which sets does the Cartesian product {(3, 3); (3, 4)} come from?

 

SOLUTIONS

Solution 1. The correct answer is A=(1, 2); B=(3, 1). To the each point in the Cartesian coordinate system an exact ordered pair (a,b) of real numbers is assigned .

Solution 2. x+y=7; x-y=1; x=1+y; 1+y+y=7; 2y=6; y=3; x=4 The correct answer is (4, 3).

Solution 3. 2 3  4 = 24

Solution 4. D E = { 1, 2, 3}; F = { 4, 5} (D E) F = 3 2 = 6  

Solution 5. f(x) = x - 2; for f(x) = 3 and x = 5. It is valid because 3 = 5 - 2, hence the correct answer is (5, 3)

Solution 6. The correct answer is {(a, b), (c, a), (b, d), (d, c), (c, a)} For S = { a, b, c, d } and f : S S.  

Solution 7. The correct answer is (a), because there is only one point is on each vertical line, meanwhile two points on the horizontal line do not spoil the properties of a function.

Solution 8. g(1) = 2; f(g(1)) = f(2) =3 The correct answer is 3.

Solution 9. {3} {3, 4} = { (3,3), (3,4) } The correct answer is {3}, {3, 4}.

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RELATIONS

Exercise 1. Let M = {1, 2, 3}, and let the relation R in M be the set of points displayed in the coordinate diagram of M × M. Which one of the following statements is true?

a) 1 R 1

b) 3 R 2

c) 2 R 2

d) 2 R 1

How many elements in M is 2 related to?

How many elements in M are related to 2?

 

Exercise 2. Two sets are given by E = {2, 4, 5, 6} and F = {1, 3, 6}. Let R be a relation from set E to set F, which is defined by the open sentence "x is greater than y". How many elements does the solution set of R contain?

 

Exercise 3. The relation R is given by the set of ordered pairs, R = {(2, 4), (3, 4), (1,3), (3, 5), (2, 3) }. Which of the following is the domain of R?

a) {2, 3, 1, 5}

b) {1, 2, 3}

c) {1, 2, 3, 4, 5}

d) {2, 4, 3, 5}

 

Exercise 4. Let R be the relation as in Exercise 3, R = {(2, 4), (3, 4), (1,3), (3, 5), (2, 3)} . What should the range of R be?

a) {3, 4, 5}

b) {1, 2, 3}

c) {1, 2, 3, 4, 5}

d) {4, 4, 2, 5}

 

Exercise 5. Find the inverse of the relation R = {(2, 4), (3, 4), (1,3), (3, 5), (2, 3)} .

a) {(2, 4), (3, 4), (1, 3), (3, 5), (2, 3)}

b) {(4, 2), (4, 3), (3, 1), (5, 3), (3, 2)}

c) {5, 4, 3, 2, 1}

d) {(4, 4), (3, 3), (1, 1), (5, 5), (2, 2)}

 

Exercise 6. Which one of the following open sentences defines a reflexive relation on the set of natural numbers?

a) "x is less than y"

b) "x = 2y"

c) "x - y = 5"

d) "x divides y"

 

Exercise 7. Find the reflexive relation on the set A if A = {a, b, c} .

a) R1 = {(a, b), (b, c), (a, a), (c, c)}

b) R2 = {(a, a), (b, c), (c, c)}

c) R3 = {(a, a), (a, c), (c, a), (b, b), (c, c)}

d) R4 = {(a, a), (c, c), (a, c), (c, a)}

 

Exercise 8. Which one of the following open sentences defines a symmetric relation in the set of natural numbers N?

a) "x is less than y"

b) "xy = 12"

c) "x - y = 5"

d) "x divides y"

 

 Exercise 9. Find non-symetric relation on the set B = {c, d, e}.

a) R1 = {(e, e)}

b) R2 = {(e, e),(c, d), (d, c)}

c) R3 = B x B

d)R4 = {(c, d), (d, d), (d, e)}

 

Exercise 10. Find the anti-symmetric relation on the set B = {1, 2, 3}.

a) R1 = {(3, 3)}

b) R2 = {(1, 2), (1, 1), (1, 3), (2, 1)}

c) R3 = B × B

d) R4 = {(1, 2), (2, 2), (2, 3), (3, 2)}

 

Exercise 11. Let C = {1, 2, 3, 4}and let R1, R2, R3, R4 be the relations in C. Which one of them is transitive?

a) R1 = {(2, 3), (3, 2), (3, 3), (1, 1)}

b) R2 = {(1, 1), (2, 2), (1, 3), (3, 2)}

c) R3 = {(3, 4), (3, 3), (4, 4), (4, 3)}

d) R4 = {(1, 2), (2, 2), (2, 3), (3, 2)}

 

Exercise 12. Find the transitive relation in the set D = {1, 2, 3, 4}.

a) R1 = {(1, 2), (4, 3), (3, 2), (2, 4)}

b) R2 = {(1, 4), (4, 2), (1, 1), (3, 2)}

c) R3 = {(3, 2), (2, 4), (4, 4), (4, 3)}

d) R4 = {(3, 1), (2, 4), (4, 3), (3, 4)}

 

SOLUTIONS

Solution 1. The correct answer is 2 R 1, because the point {2, 1} belongs to the coordinate diagram M × M. Correct answers are 2 and 1, because vertical line through 2 contains two ordered pairs in which 2 appers as first element {2,1} and {2,3}, while horizontal line through 2 contains only one point of M and that is {2, 1}.

Solution 2. The correct answer is 7. R = {(2,1), (4,1), (4,3), (5,1), (5,3), (6,1), (6,3)}

Solution 3. The set of the first elements in R defines the domain of R, therefore the domain of R is {1, 2, 3}.

Solution 4. The set of second elements in R gives the range of R, which is { 3, 4, 5 }.  

Solution 5. If we reverse the order of elements in ordered pairs that are consisted in the set that defines the relation R, we wil get the inverse relation R-1. R-1 = { (4,2), (4,3), (3,1), (5,3), (3,2)}

Solution 6. The correct answer is "x divides y" , as every natural number devides itself.

Solution 7. The relation R3 is reflexive because (1, 1), (2, 2) and (3, 3) belong to that relation.

Solution 8. The correct answer is "xy = 12", because ab = 12 implies ba = 12, for each a,bN. "x divides y" is not symmetric because "a divides b" does not imply "b divides a", for each a,b N. "x is less than y" is not symmetric: a < b does not imply b < a for each a,bN. "x - y = 5" is not symmetric because a - b = 5 does not imply b - a = 5 for each a,b N.  

Solution 9. The correct answer is R 4, because (c,d) R4 and (d,c)R4; (d,e)R4 and (e,d)R4.

Solution 10. The correct answer is R1. Other relations in the set B are not anti-symmetric because all of them fulfill the rool: if there exists elements a,bB, ab such that (a,b)R and (b,a)R then R is not anti-symmetric.

Solution 11. The correct answer is R3. Other relations in the set C are not transitive because all of them fulfill the rool: if there exists elements a,b,c C (a,b,c may be equal), such that (a,b)R and (b,c)R but (a,c)R then R is not transitive. 

Solution 12. The correct answer is R1. Other relations in the set D are not transitive because all of them fulfill the rool: if there exists elements a,b,c D (a,b,c may be equal), such that (a,b) R and (b,c) R but (a,c) R then R is not transitive.  

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