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Introductory Chapter : Mathematical Logic, Proof and Sets 1

INTRODUCTORY CHAPTER: Mathematical Logic, Proof and Sets

SECTION A Joy of Sets

By the end of this section you will be able to understand what is meant by a set understand different types of sets plot Venn diagrams of set operations carry out set operations

This section is straightforward. You will need to understand set theory notation. Once you have digested this notation then the remaining material is routine mathematical work.A1 Introduction to Set TheoryWhat does the term set mean?A set is a collection of objects and these objects are normally called elements or members of the set. The following are examples of sets:

1. The numbers 1, 2, 3 and 4.2. Students who failed the mathematics exam.3. A pack of cards.4. European capital cities.5. All the odd numbers.6. The roots of the equation .

A set can be described in various ways:i. By listing all the elements of the set. For example in 1 above the set can be

written as . The curly brackets, , capture the set and each element in the set is separated by a comma.

ii. By listing the first few elements to give an indication of the pattern of the set. For example . This is the set in number 5 above. Note that the 3 dots (ellipses), , represents the missing members when there is a pattern.

iii. By describing a property of the set such as .iv. By stating a mathematical expression like

What does the set D mean?The set D consists of the elements x such that x satisfies the quadratic equation

. The vertical line, , in the set is read as ‘such that’. Hence the set D has members x such that .In some mathematical literature the above set D is written as

The vertical line is replaced by the colon : However throughout this book we will use the vertical line in the curly brackets to represent ‘such that’.In mathematical notation sets are normally denoted by capital letters such as A, B, C … X, Y … The elements or objects of the set are denoted by lower case letters such as a, b, c … x, y …

such that


Example 1 Determine the elements of the set D given above.Solution.We need to solve the quadratic equation given in the set .We have

We can write the set D as but it can be written as . The order of the elements in a set does not matter.We denote the number 7 is a member of the set D by

The symbol means ‘is a member of’. Since 2 is not a member of this set therefore we denote this by and read it as ‘2 is not a member of the set D’.In general

means x is a member of the set AWhat does mean?

means x is not a member of the set AExample 2 Let A be the set of all even numbers. Write the set A in set notation.Solution.We can write even numbers as the symbol x such that x is an even number, thus we have

What is the size of the set A?It is an infinite set. Note that sets maybe infinite or finite. Can you think of an example of a finite set?The above set .

A2 Types of SetsThere maybe no elements in a set. What do you think we call a set which has no members?The empty set or the null set. The empty set is normally denoted by (The Greek letter phi). Can you think of any examples of the empty set?

Remember prime numbers are greater than 1.What does the universal set mean?Universal set is the set of all the elements under consideration. For example if we are discussing prime numbers then the universal set will be the set of all prime numbers. The universal set is denoted by . There are various types of numbers that we have used throughout our lives but they have not been placed in set form or been given a special symbol. Can you remember what types of numbers you have used?


Real numbers, natural numbers, rational numbers etc. We can give all of these their own symbol:

the set of all natural numbers 1, 2, 3, 4, … These are sometimes called the counting numbers.

the set of all integers … ,… This is the set of all whole numbers.

the set of all rational numbers. These are numbers which can be written as ratios

or fractions such as etc. Note that all the integers are also in

this set because numbers like 6 can be written as .

Numbers such as etc cannot be written as fractions so these are not rational numbers. These are called the irrational numbers.

the set of all real numbers. This is the set of all rational and irrational numbers.

For example 2.333 … are all members of .

the set of all complex numbers. This set contains all the real numbers as well as numbers such as which is not a real number. Complex numbers are normally written as where i denotes an imaginary number and is equal to .All the above sets are examples of infinite sets. Example 2 Determine the elements of the set .Solution.The factorized quadratic has the solutions

Does the set A contain both these elements 3 and ?

No because the set A has the qualification . What does this notation mean?x is a member of the set of natural numbers which means x is a counting number.

Since (rational) is not a natural number therefore it cannot be a member of the

set A. Thus the set A only has the element 3, that is .

The set A only has one element 3 and in general a set with only one element is called a singleton.Definition (I.1). Any set with precisely one element is called a singleton.

Example 3 Write the following statements in set notation:(a) The set of positive real numbers excluding 0.(b) The set of negative integers.(c) The set of rational numbers between 0 and 1.Solution.


(a) The set of positive real numbers can be written as a symbol x which represents a real number such that it is greater than 0:

What does in this set mean? means x is a real number.

(b) What is the symbol for the set of integers? represents the set of all integers. The set of negative integers can be written as x

which is an integer such that it is less than 0:

(c) What is the symbol for the set of rationals? represents the set of rationals (Q for quotient). The set of rationals between 0 and

1 can be written as:

A3 Venn DiagramsVenn diagrams are a graphically way of representing sets. Venn diagrams were introduced by John Venn.

Fig 1 John Venn 1834 to 1923

John Venn was born in Hull, England in 1834. His father and grandfather were priests and John was also groomed for a similar post. In 1853 he went to Gonville and Caius College Cambridge and graduated in 1857 becoming fellow of the college. For the next 5 years he went into priesthood and returned to Cambridge in 1862 to teach logic and probability theory. John Venn is popular known as the person who developed a graphically way to look at sets and this graph become known as a Venn diagram. The sets were represented by oval or circular shape figures but they can be any shape.

http://en.wikipedia.org/wiki/Image:John_Venn.jpg


In 1883 John Venn was elected as a Fellow of the prestigious Royal Society. At this time he started to take an interest in history and by 1897 he published a history of his college Gonville and Caius, Cambridge. Consider the set . What are the elements of the set A?A is the set of integers which lie between to 2. Thus the elements are

and 2. A Venn diagram of this looks like:

Fig 2The U in the bottom right hand corner of the rectangle is the universal set which means it includes every element under consideration. The members of the set A lie within the boundary of the oval shape as shown in Fig 2. We can use Venn diagrams to display set operations.A4 Union and Intersection of SetsFrom the age of 5 we have added and subtracted numbers. In a similar fashion we can carry out similar operations on sets. These operations are called union and intersection. What is the union of two sets?The word union in everyday language means combining of 2 or more things. Union of two sets is the combination of all elements in both sets.Definition (I.2). The union of two sets A and B is the set of all the elements belonging to set A or set B. The union of two sets A and B is denoted in set theory notation as

and

In terms of a Venn diagram we can draw this as:

Fig 3 (A union B) is shadedThe other operation on sets is intersection. What does intersection mean in everyday language?Intersection means crossroads. Intersection of two sets A and B is the set of all elements which belong to both sets A and B. Definition (I.3). The intersection of two sets A and B is the set of all the elements belonging to set A and set B. The intersection of two sets A and B is denoted in set theory notation as and

The Venn diagram of is:


Fig 4 (A intersection B) is shaded

Example 4 Let and . Determine the sets (A union B) and (A intersection B).Also draw the Venn diagrams of these sets.Solution.What does mean?A union B is the set of all elements which are in the set A or B. Thus we have

Which elements does the set have?is the set of all elements which belong to both the sets A and B. Which elements

are common to both the sets and ?Only the number 3 belongs to both sets A and B. Therefore

Note that is a singleton.The Venn diagrams of and are

Fig 5 is shaded is shaded

Example 5 Let and . Determine the sets and . Draw Venn diagrams of and .Solution.What does the notation mean?

is the set of all the even and odd numbers which means it is the set of all the integers. Which symbol is used to represent all integers?

is the set of all integers. Thus we have . What is equal to?There is no element which is common to both the set of even and odd numbers. Therefore the intersection of these sets is empty. What is the symbol for the empty set?The Greek letter phi, , denotes the empty set. Thus we have

Venn diagrams of these is:

is shaded

O E


Fig 6In general if for given sets A and B we have then we say the sets A and B are disjoint.We can extend the above set operations to 3 or more sets.Example 6 Let , and . Determine the elements of the following sets:(a) (b) What do you notice about your results?Draw a Venn diagram of the set .Solution.(a) How do we find the elements of ?First we determine . Thus is the set of elements which are common to both sets and . That is

is the set of elements which are in set or set

. Which elements are in either of these sets?It is all the elements in set A and the element 9:

(b) How do we find the elements of ?Well is the set of elements which belong to either of these given sets

or :

Similarly the elements in set or are

What does mean?It is the set of elements which are common to both and . Which elements belong to both these sets and

?

Note that the answers to parts (a) and (b) are the same, that is

Venn diagrams of are


Fig 7Shading in the union of these sets of Fig 7 gives:

Fig 8

The last observation, , is true for all sets and we will prove this result in later sections. You may like to convince yourself of this result by drawing general Venn diagrams.A5 Other Set OperationsWhat does the word complement mean in everyday language?Complement is something which completes or fills up. In set theory the complement of a set A is the elements which are in the universal set but not in set A.Definition (I.4). The complement of a set A is denoted by and is defined to be

What does the Venn diagram of look like?

(complement of A) is shaded Fig 9You may see written as in other mathematical literature.

Example 7 Let and universal set . Determine .Solution.What does mean?The universal set is the set of natural numbers 1, 2, 3, 4 … Note that E is the set of even numbers. What does mean?

is the set of elements which are not in the set E but are in the universal set. This means numbers which are not even but are natural numbers. What numbers are these?The odd numbers. Thus .

In mathematics what does difference mean?Difference is subtraction. Given two sets A and B the difference of A and B is the set of elements which belong to the set A but not to B. [Subtract the members of B from A].


Definition (I.5). The difference of set A and set B is denoted by and is defined to be

What does the Venn diagram of look like?

Fig 10 is shadedIn other mathematics literature you might find the following notation for difference of two sets such as or .Example 8 Let and be two given sets. Find . Solution.What does the set theory notation mean?From the set A subtract the members of set B. This means from the set

we delete members which are in the set and the remaining set is . Thus

There is one more difference in set theory called symmetric difference. The symmetric difference between two sets A and B are the elements which are in set A or set B but not in both.Definition (I.6). The symmetric difference of set A and set B is denoted by and is defined to be

What does the Venn diagram of this look like?

Fig 9 is shadedIn the exercises we will show the result . This means

is the set of take away the elements of .Example 9 Let , and . Find , and

. Solution.The elements in are members which are in the set A or B but not in both. Which elements are in both sets and ?


The elements 1 and 3 are common to both sets A and B. Thus removing elements 1 and 3 from the union of sets A and B gives

Similarly we can find . What members are common between the sets and ?

1 and 4 therefore removing these from gives

To find we delete the members which are in common between the sets and . Only the element 1 is common therefore

Example 10 For the Venn diagram of Fig 9 above show that

Solution.We first shade in and then which is everything outside :

Fig 10Similarly for we shade in and and then take the union of these 2 sets,

.

Fig 11Notice that the same regions are shaded in Fig 10 and 11 for and .

We have for the sets shown.

You can show results of set theory by using Venn diagrams like these but they are only valid for the sets shown. We will prove results of set theory more rigorously in later sections by using algebra of subsets.

SUMMARYA set is a collection of objects. Capital letters are used to represent sets and lower case letters for elements of sets. The notation means the element x is a member of the set A.There are various types of sets such as the empty set , the universal set U and the sets which give the type of numbers:

Natural numbers. These are positive whole numbers.Integers. These are whole numbers.Rational Numbers. These are ratios of integers.


Real numbers. These are rational as well as irrational numbers. Complex numbers. These are numbers of the form .

Any set with just one element is called a singleton.Let A and B be sets. Then we have the following set operations:The union of two sets is given by

The intersection of two sets is given by

The complement of a set A is given by .

The difference of sets A and B is given by The symmetric difference of sets A and B is given by

introductory chapter: mathematical logic, proof and setsmaths-for-all.co.uk/userfiles/section a 18th...

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