introduction. sets · the following venn diagram illustrates the relationship among the three sets...

Szent Istvan University, Faculty of Veterinary ScienceDepartment of Biomathematics and Informatics

Biomathematics 1

Introduction. SetsJanos Fodor

Copyright c© [email protected] Revision Date: September 5, 2006 Version 1.25

mailto:[email protected]

Table of Contents

0 What is biomathematics? 3

0.1 Why to study biomathematics? . . . . . . . . . . . . . . . . . 3

0.2 Key to success . . . . . . . . . . . . . . . . . . . . . . . . . . 3

0.3 Course outline . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.4 Requirements and Grading . . . . . . . . . . . . . . . . . . . . 4

0.5 Course Home Page . . . . . . . . . . . . . . . . . . . . . . . . 5

0.6 Useful Literature . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Sets 6

1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 The Set of Real Numbers . . . . . . . . . . . . . . . . . . . . 7

1.3 Describing Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Equality and Subsets of Sets. The Empty Set. . . . . . . . . . 10

1.5 Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Set Operations 12

2.1 The Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The Intersection . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Set Difference. Complement of a Set . . . . . . . . . . . . . . 16

Section 0: What is biomathematics? 3

0. What is biomathematics?

Biomathematics is the branch of mathematics applied to living systems andprocesses.

Mathematics provides methods applicable in other disciplines. We are going tofind mathematical methods to answer certain questions of life sciences.

We discuss the details of the applications rather than the details of the math-ematical machinery.

0.1. Why to study biomathematics?

The aim is to prepare you for an understanding of the basic mathematicalmodels and methods that are useful in your major field.

Life scientists are hardly interested in going deeply into mathematics. Therefore,concepts are introduced in an intuitive way. The relevance of the procedures isproven by examples that have been selected from a wide area of life sciences.The course uses a common-sense approach to explain basic ideas and methods.Real-life examples show how each idea or method is applied in practice.

As the course will show you very soon, mathematics is useful in understandingthe world around you, and it is relevant to your main interest. We think theresult is a friendly and informal survey that will help you long after the courseis over.

0.2. Key to success

The key to success is to take a

positive attitude

and invest a reasonable amount of

time and effort.

Everyone is sometimes frustrated when trying to learn new material. But therewards, in understanding the world and gaining new abilities – plus enjoymentand a sense of accomplishment – will make it worthwhile.


0.3. Course outline

Main topics of the course:

1. Sets

2. Matrices

3. Functions and their applications

4. The derivative and its applications

5. The integral and its applications

6. Ordinary differential equations and their applications

7. Population dynamics

8. Linear programming

9. Probability

10. Descriptive statistics

11. Estimation

12. Hypothesis testing

13. Linear regression and correlation

14. The analysis of variance

0.4. Requirements and Grading

Regular attendance (including punctual arrival in time for the scheduled begin-ning) of practicals, according to the actual group assignment, is an absoluterequirement.

There are two midterms, 25 points each. The sum of the two midtermpoints must be at least 25 in order to be allowed to take the final exam(written).


Midterm tests can be repeated at most once, only if you have not missed morethan two practicals. Detailed information is provided by practical teachers intime.

The final exam (written) is held in the examination period, it’s value is 50points. It is based exclusively on the lectures.

Therefore, you can have a maximum of 100 points total at the end.

The final grade is computed on the basis of the following table:

Total point Final grade

0 – 50 151 – 63 264 – 76 377 – 90 491 – 100 5

0.5. Course Home Page

From the next Monday, the home page of the course can be found athttp://www.univet.hu/users/jfodor/biomathematics.htm

You can download all the lectures as PDF files, in two versions. One for readingit on your computer screen, one for printing. To read or print these files, youneed the Acrobat Reader (freeware).

In these files the following conventional notations (colors) are used:

DEFINITION (introduction of a new notion)

THEOREM (a provably true statement)

Section 1: Sets 6

0.6. Useful Literature

1. J. Fodor, Biomathematics – Part One, (available at our Department;price: 1100 HUF).

2. J. Fodor, Biomathematics – Part Two, (available at our Department;price: 700 HUF).

3. L.D. Hoffmann and G.L. Bradley, Calculus for Business, Economics and theSocial and Life sciences. Seventh edition, McGraw-Hill, 2000.

Online version is available free of charge (see the link at the course home-page).

4. A.G. Bluman, Elementary Statistics: A Step By Step Approach. Fourthedition, McGraw-Hill, 2001.

Online version is available free of charge (see the link at the course home-page).

1. Sets

Everyday language contains several words to designate a collection of objects:

• order, family, genus (for animals and plants with certain characteristics incommon)

• social class, sample, group, category, etc.

In mathematics we prefer the term set.

1.1. Basic Notions

A set is a collection of objects with the important property that we can tellwhether any given object is or is not in the set. Ambiguity is not allowed.

Example.

• The set of all plants that produce O2 is well-defined, since these plantscontain chlorophyll.

• The group of broad-leaved plants is not a set. The judgment of “broad-leaved” is subjective and causes ambiguity.

Section 1: Sets 7

Each object in a set is called an element of the set.

Symbolically,

a ∈ A means a is an element of set A (a belongs to A);

a /∈ A means a is not an element of set A (a does not belong to A).

1.2. The Set of Real Numbers

Informally, a real number is any number that has a decimal representation.

• N: the set of natural numbers (positive integers)

• N0: the set of nonnegative integers

• Z: the set of integer numbers (positive, negative, zero)

• Q: the set of rational numbers (quotients of two integers; decimal repre-sentations are repeating or terminating:

−4, 0, 1.25, −35 , 2

3 , 3.272727, 0.666)

• Q∗: the set of irrational numbers (numbers that can be represented asnonrepeating and nonterminating decimal numbers:√

2 ≈ 1.414213 . . ., π ≈ 3.14 . . ., 7√

3, e ≈ 2.71828182 . . . )

• R: the set of real numbers (rational numbers and irrational numbers).

For example, 1 ∈ N; 0 /∈ N;√

2 /∈ Q; −1/2 ∈ Q.

Geometric interpretation of RConsider an arbitrary straight line L, and choose two distinct points on it: oneof which we identify with (or assign to) the number 0, and the other (whichlies to the right of 0) with the number 1.

Section 1: Sets 8

The scale on L: the unit of distance is the line segment between points 0 and1.

Then for any two numbers a and b, a < b if and only if a lies to the left of b.

The line which has been identified with R under the correspondence we justdescribed is called a real number line.

Example. (Intervals.) Let a < b real numbers. There are four different kindsof bounded intervals with endpoints a, b:

• [a, b] := {x | a ≤ x ≤ b} (closed interval);

• ]a, b[ := {x | a < x < b} (open interval);

• ]a, b] := {x | a < x ≤ b} ;

• [a, b[ := {x | a ≤ x < b} ;

• ]−∞, a] := {x | x ≤ a} ;

Caution: infinity (denoted by the symbol ∞) is NOT A NUMBER (it isnot a point on the number line), and should not be used like one. In the intervalnotation it indicates unbounded subsets of R.

Section 1: Sets 9

1.3. Describing Sets

A set is called finite if it contains only a finite number of elements; otherwiseit is called infinite.

A set is usually described in one of two ways:

• by listing the elements between braces {} (when the set is finite), or

• by enclosing within braces a rule that determines its elements.

For example,

{sight, hearing, smell, taste, touch}

is the set of the traditional five senses.

If we want to denote this set by S, we use the symbol := as follows:

S︸︷︷︸new symbol

:= {sight, hearing, smell, taste, touch}︸︷︷︸already known notion

It means that S, by definition, is equal to the set of five senses.

Example. If D is the set of all numbers x such that x2 = 4, then using thelisting method we write

D = {2,−2} (listing method),

or, using the rule method we write

D = {x ∈ R | x2 = 4} (rule method).

Note that in the rule method, the vertical bar represents “such that,” and theentire symbolic form {x ∈ R | x2 = 4} is read:

“The set of all real numbers x such that x2 = 4.”

The letter x introduced here is called a variable.

A variable is a symbol that is used as a placeholder for the elements of aset.

Section 1: Sets 10

A constant, on the other hand, is a symbol that names exactly one object.

The symbol “8” is a constant, since it always names the number eight.

Example.

X := {x ∈ R | x22 − 9x9 + 3x4 + 4x + 1 = 0}.

Can we list the elements of X explicitly? Hardly.

Can we decide for any given number if it belongs to X? Yes, we can. So, X isindeed a set.

1.4. Equality and Subsets of Sets. The Empty Set.

Two sets A and B are equal if they contain exactly the same elements. Insymbols: A = B.

Example.

{−2, 5} = {5,−2} (the order of elements does not matter);

{1, 5} = {1, 5, 5, 1, 5} (repeating elements does not matter).

Consider the set of all people who have walked on the sun. There is no suchperson, so this set contains no elements.

Let X := {x ∈ R | x2 = −1}. This set contains no elements!

A set is called empty if it contains no elements.

There is only one empty set, it is denoted by ∅.

If each element of set A is also an element of set B, we say that A is asubset of B, and we write A ⊆ B.

Note that

• every set is a subset of itself

(for example, {1, 5, 3} ⊆ {1, 3, 5});

Section 1: Sets 11

• the empty set is a subset of every set;

• for any two sets A and B we have

A = B if and only if A ⊆ B and B ⊆ A .

Example. {1, 5} ⊆ {1, 3, 5}, but {2, 5} is not a subset of {1, 3, 5}

If A is not a subset of B, we write A * B.

If A ⊆ B and A 6= B then A is called a proper subset of B, and we writeA ⊂ B.

Example. {1, 5} ⊂ {1, 3, 5}; N ⊂ Z ⊂ R;

{−1,√

2} ⊂ R; {−1,√

2} 6⊂ Q; {1, 2, 3} ⊆ {1, 2, 3}.

1.5. Venn Diagrams

Set relationships can often be illustrated by a device known as a Venn diagram.

Sets of any kind of members are represented by sets of points on the plain(points in a circle or in a rectangle).

For instance, the next figure shows a Venn diagram for the simple inclusionB ⊂ A.

Venn diagram for representing B ⊆ A.

B

A

Section 2: Set Operations 12

Neither A nor B is a subset of the other:

The following Venn diagram illustrates the relationship among the three sets

{March},{March, April, May}, and{June, July, August}:

2. Set Operations

Sets can be combined in several different ways to form new sets.

The set operations we shall consider are

• union,

• intersection,

• difference, and as a particular case,

• complementation.


2.1. The Union

The union of two sets A and B is the set of all elements that are elementsof A or of B or of both, and is denoted by A ∪B. That is,

A ∪B := {x | x ∈ A or x ∈ B}.

Example. {−5, 0, 2} ∪ {0, 3} = {−5, 0, 2, 3}

We can picture the union in the Venn diagram shown below:

Properties of the union:

Let A, B, C be sets. Then we have

1. A ∪∅ = A , A ∪ A = A ;

2. A ∪B = B ∪ A (commutativity);

3. (A ∪B) ∪ C = A ∪ (B ∪ C)(associativity).

Hence, the union of more than two sets does not depend on the order of sets:

A ∪B ∪ C := (A ∪B) ∪ C.

For example,

{4, 2} ∪ {2} ∪ {−1,−2,−3} ∪ {1,−1,−2} =

= {−3,−2,−1, 1, 2, 4}.

2.2. The Intersection

The intersection of two sets A and B is the set of all elements belong toboth A and B (common elements of both sets). It is denoted by A∩B, and

A ∩B := {x | x ∈ A and x ∈ B}.

We can picture the intersection in the Venn diagram shown below:


Example. {−5, 0, 2} ∩ {0, 3} = {0};

{1, 2} ∩ {−1, 3} = ∅.

We say that sets A and B are disjoint if A ∩B = ∅.

Properties of the intersection:

Let A, B, C be sets. Then we have

1. A ∩∅ = ∅ , A ∩ A = A ;

2. A ∩B = B ∩ A (commutativity);

3. (A ∩B) ∩ C = A ∩ (B ∩ C)(associativity).

The intersection of more than two sets does not depend on the order of thesets: A ∩B ∩ C := (A ∩B) ∩ C.

For example, {4, 2} ∩ N ∩ {1,−1, 2,−2} ∩ R = {2}.

Some problems involving both union and intersection can be simplified by useof distributive laws, which are similar to those used for addition and multipli-cation of numbers.

For any sets A, B, C we have

1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)(the intersection is distributive with respect to the union);

2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)(the union is distributive with respect to the intersection).

Pictorial “proof” of the second property:


Green shaded area: A ∪B

If you try to solve the following problem without using the distributive law, youwill find that the sets you have to work with are more complicated.


Example. Define the sets

A := {x ∈ R | x is a positive integer less than 14 },

B := {x ∈ R | x is odd}, and

C := {x ∈ R | x is a positive multiple of 3}.

Use the first distributive law to obtain A ∩ (B ∪ C).

Solution. Since

A ∩B = {1, 3, 5, 7, 9, 11, 13} and

A ∩ C = {3, 6, 9, 12}, we find that

A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)

= {1, 3, 5, 6, 7, 9, 11, 12, 13}.

2.3. Set Difference. Complement of a Set

Let A and B be sets. The difference of A and B is the set that containsthose elements of A which are not in B. We denote the difference of A andB by A \B. Thus,

A \B := {x | x ∈ A and x /∈ B}.

For instance, {2, 3, 4} \ {2, 5} = {3, 4} and

{2, 5} \ {2, 3, 4} = {5}.

We can visualize set difference by the Venn diagram shown in the next figure.

In the next case we fix a universal set S and consider only subsets A of S. Thenelements of S which are not in A form the complement of A in S.


Let S be a fixed set (called the universal set), and A be any subset of S.The complement of A in S is the set of all elements in the universal set S

that are not elements of A. We denote the complement of A by A. Thus,

A := S \ A.

For example, if S denotes the set of students in the first year at our faculty,and A is the set of those ones who are sitting in the lecture room now, thenS \ A is the set of students doing anything else but being present.

We can visualize the complement of a set by the Venn diagram shown in thenext figure.

The following two rules, called De Morgan’s laws, provide useful formulas forthe complements of unions and intersections.

(De Morgan’s Laws)

For any sets A and B we have

(A ∪B) = A ∩B , and (A ∩B) = A ∪B .

In words: the complement of the union is the intersection of the complements,and the complement of the intersection is the union of the complements.

Illustration


Grey area: (A ∪B)

Outside green: A

Outside red: B

Try solving the following example directly, without using De Morgan’s laws.

Example. Farmer Jones’ chickens are either speckled or brown. Some are bigand the others are small. Let

S := {x | x is a chicken owned by Farmer Jones},

A := {x | x is a speckled chicken}, and

B := {x | x is a big chicken}.

What does the set A ∪B represent?

Solution. Since

A = {x | x is a brown chicken} and

B = {x | x is a small chicken},

it follows from De Morgan’s laws that

A ∪B = A ∩B = {x | x is a small, brown chicken}.

introduction. sets · the following venn diagram illustrates the relationship among the three sets...

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