UNIT 4 – SETS

Exercises:   [1]   [2]   [3]   [4]   [5]   [6]   [7]   [8]

INTRODUCTION

Mathematics is traditionally thought of as dealing with numbers, that is quantities like distances, angles, weights etc., but it also includes non-numerical branches such as logic. These areas have developed because the study of most disciplines is facilitated by a precise and symbolic method of expression.

The English language contains many words to designated a collection of objects. Biologists use categories such as order, family, and genus to mean collections of plants and animals which have certain characteristics in common. Sociologists subdivide the population of a country into social classes; economists subdivide it into income groups. When statisticians select individuals form a population they use the word sample. All these words collection, class, group, category, etc. have some common meaning mathematician use the term set for it.

This unit is an introduction to the language and concepts used in mathematics for discussing sets, parts of sets, combination of sets, etc.

OBJECTIVES

1. To describe a set using two standard forms of notation.

2. To perform operations on sets.

3. To use Venn diagrams to illustrate and solve problems involving sets.

SETS

A set is a well-defined collection of any kind of objects: people, ideas, plants, numbers, responses, etc. It must be well-defined in the sense that there must be no ambiguity about whether an object belongs to a set or not. For example, it is easy to define, unambiguously, the set of patients in a particular hospital at a particular time, but you cannot define the set of all broad-leafed plants because the judgment of broad-leafed is subjective and might cause ambiguity.

Exercises 4.1 Solutions

Which of the following collections qualify as sets according to the rigorous definition above?

1. All firms made in Australia.

2. The chemical elements.

3. All great Australian novels.

4. The integers.

5. All species in the family Eucalyptus.

ELEMENTS

The objects belonging to a set are called its members or elements. The symbol e is used to denote ‘is a member of’ or ‘belongs to’. If a set is denoted by T then, a e T means a is an element of T, and b Ï T means b is not an element of T’.

The number of elements is a set may be finite or infinite. For example the set of cabinet members contains a finite number of elements whereas the set of all living cells is infinite.

Exercises 4.2 Solutions

State whether the collections which qualified as sets in Exercises 4.1 are finite or infinite.

NULL SET

This is a set which has no members. It is also called the empty set and is denoted by ø. Surprisingly, perhaps, this is a useful concept; e.g. the collection of all your living aunts is a set, but for some people this set may be empty. Another empty set is the set of female prime ministers of Australia before 1979.

NOTATION FOR SETS

A set is described in one of the following ways.

1. Listing all its elements,

   e.g. S is the set of the five senses,

   S = {sight, hearing, smell, touch, taste}

   e.g. A is the set of natural numbers less than 5,

   A = {1, 2, 3, 4}

   e.g. E is the set of all positive even integers,

   E = {2, 4, 5, …}.

   Note that the order in which the elements are written is not important,

   e.g. the set of natural numbers less than 5 can be written as

   A = {2, 4, 1, 3}

2. Writing down the property which characterises the elements of the set,

The symbol | which is read as such that is sometimes replaced by : or ; but they all have the same meaning,

Exercises 4.3 Solutions

1. List all the elements of the following sets:

   (a) A = {x|x is a natural number less than 10}

   (b) B = {s;s is a car which can travel at the speed of light}

2. Write each of the following sets in the standard notation using {} and, if you are describing the common property of the elements, using the symbol | (or ; or : ).

1. The set of integers between -5.1 and 2.1.

2. The set containing all animals that have pouches to carry their young.

3. The set consisting of natural numbers less than 20 and divisible by 3.

1. If A = {1, 2, 3, 4, …, 20} indicate whether the following statements are true or false:

   (a) 10 e A (b) 4 Ï A (c) 25 e A

   (d) 3.5 e A (e) 0 e A (f) 6½ Ï A

2. Write each of the following sets in standard notation, first by listing its elements and then by using the property which characterises its elements.

1. A is the set of natural numbers that are multiples of 5.

2. B is the set of fractions between (but not including) –1 and 1 that have 3 as a denominator.

3. C is the set of all countries located on the Australian continent.

EQUALITY OF SETS

Two sets A and B are said to be equal, in symbols

A = B

If they contain exactly the same members.

e.g. {1, 2, 3] and {3, 1, 2} have the same members of they are equal.

Also {2, 4, 6} = {6, 4, 6, 2, 4} for the same reason.

SUBSETS

A set P is called a subset of set Q if all the elements of P are also elements of Q. This definition includes the case of equality (that is , if P = Q, P is also a subset of Q). Symbolically ‘P is a subset of Q’ is written as:

P Í Q

The fact that equality is included in the above definition can sometimes be inconvenient. To overcome this, P is defined to be a proper subset of Q if there are some elements in Q which are not in P, (i.e. P does not equal to Q). This is written as:

P Ì Q

Notice that the symbols Ì and Í are analogous to the inequality symbols < and (recall Unit 2).

Example 4a. If A = {x : x is an animal}

B = {y : y is an animal with a tail}

C = {horses, kangaroos}

and D = {cows, guinea pigs, horses, kangaroos}

Which of the following statements are true?

Exercises 4.4 Solutions

1. If A = {q, r, s}, B = {p, q, r, s, u, v} and D = {u, v} are the following statements true or false?

   (a) A Ì B (b) B Í A (c) D Ì A

   (d) D Ì B (e) A É D (f) A = B

2. Indicate whether the following statements are true or false if A = {1, 2, 3, 4}, B = {1, 2}, D = {3, 4, 5} and E = {3, 4, 5}

   (a) B Ì A (b) D Í E (c) D = E

   (d) 4 Î E (e) E Ì B (f) 4 Ï B

3. Which, if any, of the following sets are equal?

B = {x ; x is a four-sided triangle}

C = {-1, 0, 1, 2, 3}

D = {n ; n is a natural number and n < -1}

VENN DIAGRAM

A useful way of depicting sets and their relationships was devised by the English logician and theologian Johan Venn (1834-1923). In a Venn diagram sets are represented graphically by circles or rectangles (or any other suitable shape). For example the Venn diagram in Figure 4.1 illustrates the relationship P is a proper subset of Q

Figure 4.1: P Ì Q

Similarly the Venn diagram in Figure 4.2 shows that A and B are two sets which share some common elements (represented by the shaded area where A and B overlap).

Figure 4.2

UNIVERSAL SET

The set of all individuals in which you are interested is called the universal set and is usually denoted by U or W (Greek letter capital omega).

By convention, it is represented in a Venn diagram as the outer boundary containing all the elements of interest. There is no prescribed shape for this boundary however a rectangle is frequently used for convenience.

COMPLEMENTARY SET

The complement of a set A is the set of all those elements of the universal set which are not in A . It is written as

or sometimes as A' or W - A. It is the shaded area in Figure 4.3.

Figure 4.3: A and

Exercises 4.5 Solutions

1. Consider all dogs as the universal set U. Purebred labradors constitute a subset L of U. What does the subset consist of?

2. If U = {2, 4, 6, 8, 10, } and A = {2, 4, 6} what is ?

UNION

The union of two sets A and B is the set which contains all the elements of A and all the elements of B (and hence all the elements which are in both A and B). It is written as A È B and illustrated by the Venn diagram in Figure 4.4.

Figure 4.4: A È B

e.g. If A = {x, y, z, p, q} and B = {a, p, z}

then A È B = {x, y, z, p, q, a}

In symbols the union of A and B is written as

A È B = {x|x e A or x e B}.

Notice that the world or is used in mathematics in the sense of and/or. Thus the statement 2 + 2 = 4 or pigs can fly is true, so is pigs are mammals or quadrupeds but 2 + 2 = 5 or elephants can fly is false.

Exercises 4.6 Solutions

1. If A = {1, 4, 7}, B = {1, 2, 5, 7}, D = {5, 6, 8, 7} and E ={9, 10} find the following:

   (a) A È D (b) B È D (c) A È (B È D)

   (d) A È B È D È E (e) E È (A È D)

2. Suppose that V = {tomatoes, potatoes, carrots} and S = {yams, lettuce, potatoes, celery, pumpkin}. If R is the set of all root vegetables, is R a subset of V È S ?

3. Are the following statements true or false?

1. 2 + 2 = 4 or 3 + 1 = 5.

2. It is not raining or it is raining.

3. People have 10 fingers or people have 10 toes.

4. 2² ¹ 4 or 16^-¼ = 2.

INTERSECTION

The intersection of two sets A and B is the set whose elements are common to both A and B. It is written as

A Ç B

And illustrated by the Venn diagram in Figure 4.5

e.g. If A = {a, b, c, d, x, y, z}

and B = {p, q, d, x, t, a, y, m}

then A Ç B = {d, x, a, y}

Figure 4.5: A Ç B

If A and B have no common elements, that is, their intersection is the empty set

A Ç B = Ø,

Then they are said to be disjoint sets (see Figure 4.6).

Figure 4.6: A and B are disjoint

Exercises 4.7 Solutions

1. Let A = {1, 2, 3, 4}, B = {2, 3, 4, 5} and C = {2, 3}.

1. What is A Ç B ?

2. Are B and C disjoint?

1. If C = {v, w, x}, D = {x, y, z} and E ={w} what are the following?

   (a) C È D (b) D È E (c) C È E

   (d) C Ç D (e) C Ç E (f) D Ç E

   (g) C È ø (h) C Ç ø

2. If A = {1, 4, 7}, B = {1, 2, 5, 7}, D = {5, 6, 8, 9} and E = {9, 10} find the following:

1. A È (B Ç D) (b) B È (E È A)

1. Suppose each individual in a population has the antigen M or the antigen N or both of them. Use set notation to describe the subset of all individuals having both M and N.

VENN DIAGRAMS AND PROBLEM SOLVING

The language of sets and, more particularly, Venn diagrams can be used to analyse and solve problems in many practical situations.

The first and most important step is to identify the sets involved in the problem. The process of defining appropriate sets requires clear thinking; this can be acquired with practice and is an essential skill for problem-solving scientists. In the following worked examples relevant sets are defined and used in the solution of the problem. You should try to identify sets in problems you meet in other areas of your study.

The shaded are M Ç F Ç S in Figure 4.7 represents the area containing the houses which would be suitable for all members of the family.

1. with respect to the presence (Rh+) or absence (Rh-) of the Rh antigen, and

2. if neither of the antigens A or B is present he has blood group 0, if only A is present he belongs to group A, if only B is present he belongs to group B, and of both A and B are present he has blood group AB.

1. Use a Venn diagram to illustrate all the possible blood types that may occur.

2. Express these blood types as intersections of sets.

Description	Symbol	Number
Women not taking history	W È	12
Women taking history	W Ç H	40 – 12 = 28
Men taking history	Ç H	73 – 28 = 45
Men not taking history	Ç	(100 – 40) – 45 = 15

Figure 4.10

Exercises 4.8 Solutions

1. A high school class consists of 60 students. Each student may take one or more of the three sports, swimming, squash and karate, if he wishes. 5 students take no sport at all. Of the 31 who attend swimming 14 take no other sport and 5 take both karate and squash as well. 10 students take only squash and a further 23 take squash plus one or more other sports. Karate as a single sport is taken by 6 students. How many students take both karate and swimming but not squash?

2. Describe the shaded area in the Venn diagram in Figure 4.11 in terms of unions, intersections and complements in two different ways.

    

   Figure 4.11

3. Using Venn diagrams show that

   (AÈ B) Ç () = ( A Ç ) È ( Ç B)

4. The following results were reported by a market researcher from interviews with 100 people:

1. all three types of washing powders,
2. only washing powder J.