Introductory Chapter : Mathematical Logic, Proof and Sets 10
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 Theory
What 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:
- The numbers 1, 2, 3 and 4.
- Students who failed the mathematics exam.
- A pack of cards.
- European capital cities.
- All the odd numbers.
- 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 …
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.
Hence the set . The order of the members of the set does not matter, that is
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 A
What does mean?
means x is not a member of the set A
Example 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 Sets
There 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 Diagrams
Venn diagrams are a graphically way of representing sets. Venn diagrams were introduced by John Venn.
Fig 1 John Venn
1834 to 1923
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 2
The 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 Sets
From 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 shaded
The 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
Fig 6
In 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 7
Shading 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 Operations
What 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 9
You 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?