3- Set:

Chapter introduction:

The theory of sets is the fundamental system of mathematics on which all other mathematical theory is founded. The first aspect of set theory that one encounters is the friendly face of intuitive or naive set theory.

3.1 Introduction to Set Theory:

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

The elements set, also called its members, can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. The statement that sets A and B are equal means that they have precisely the same members (i.e., every member of A is also a member of B and vice versa).

Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple.

Definition:

If A is a set and x is an object that belongs to A, we say that x is an element of A or that x is a member of A, or that x belongs to A, and write x A.

For example, if A is the set of all even natural numbers, then 10 A but 5 A (5 is not an element of A)

Example: Set A is {1,2,3}. You can see that 1 A, but 5 A

* Note:You simply list each element, separated by a comma, and then put some curly brackets around the whole thing.

The curly brackets { } are sometimes called "set brackets" or "braces".

The three dots ... are called an ellipsis, and mean "continue on".

There are two sets:

The first set (with the "...") we call an infinite set, The second set we call a finite set.

But sometimes the "..." can be used in the middle to save writing long lists:

Example: the set of letters: {a, b, c, ..., x, y, z}

In this case it is a finite set (there are only 26 letters, right?)

Some More Notation

When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set.

So for example, A is a set, and a is an element in A. Same with B and b, and C and c.

Numerical SetsSo what does this have to do with math? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}Set of odd numbers: {..., -3, -1, 1, 3, ...}Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}Positive multiples of 3 that are less than 10: {3, 6, 9}

And the list goes on. We can come up with all different types of sets.

There can also be sets of numbers that have no common property, they are just defined that way. For example:

{2, 3, 6, 828, 3839, 8827}{4, 5, 6, 10, 21}{2, 949, 48282, 42882959, 119484203}

3.2 Universal and Empty set

At the start we used the word "things" in quotes. We call this the universal set and is denoted by the symbol U. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to the problem you have.

There is a set with no members , which is called the empty set (or the null set) and is denoted by the symbol ∅ (other notations are used; see empty set).

Subsets:

Definition:

If A and B are sets, A is said to be a subset of B, written A B provided every member of B is a member of A. sets A and B are said to be equal, written A=B, provided A B and B A. If B is a subset of A and A B, then B is said to be proper subset of A.

For example, we have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:

A is a subset of B if and only if every element in A is in B.

So let's use this definition in some examples.

Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}? 1 is in A, and 1 is in B as well. So far so good. 3 is in A and 3 is also in B. 4 is in A, and 4 is in B.That's all the elements of A, and every single one is in B, so we're done.Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

So for example, {1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

On the contrary, {1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

You should notice that if A is a proper subset of B, then it must also be that A is a subset of B.

*Note:

When we say that A is a subset of B, we write A B.

Or we can say that A is not a subset of B by A B ("A is not a subset of B")

When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B.

3.3 Set Operations

Unions

There are ways to construct new sets from existing ones. Two sets can be "added" together. The union of A and B, denoted by A   B, is the set of all things which are members of either A or B.

The union of A and B, or A B

Examples:

{1, 2} {red, white} = {1, 2, red, white}. {1, 2, green} {red, white, green} = {1, 2, red, white, green}. {1, 2} {1, 2} = {1, 2}.

Some basic properties of unions are:

A B = B A. A (B C) = (A B) C. A (A B). A A = A. A ∅ = A. A B if and only if A B = B.

Intersections

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.

The intersection of A and B, or A ∩ B.

Examples:

{1, 2} ∩ {red, white} = ∅. {1, 2, green} ∩ {red, white, green} = {green}. {1, 2} ∩ {1, 2} = {1, 2}.

Some basic properties of intersections:

A ∩ B = B ∩ A. A ∩ (B ∩ C) = (A ∩ B) ∩ C. A ∩ B A. A ∩ A = A. A ∩ ∅ = ∅. A B if and only if A ∩ B = A.

Complements

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B \A, (or B −A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.

The relative complementof A in B.

The complement of A in U.

Examples:

{1, 2} \ {red, white} = {1, 2}. {1, 2, green} \ {red, white, green} = {1, 2}. {1, 2} \ {1, 2} = ∅. {1, 2, 3, 4} \ {1, 3} = {2, 4}. If U is the set of integers, E is the set of even integers, and O is the set

of odd integers, then the complement of E in U is O, or equivalently, E′ = O.

Some basic properties of complements:

A A′ = U. A ∩ A′ = ∅. (A′)′ = A. A \ A = ∅. U′ = ∅ and ∅′ = U. A \ B = A ∩ B′.

3.4 Set of real numbers, Subsets

Q: What is the simplest idea of a number?

The Counting Numbers or Natural Numbers We can use numbers to count: 1, 2, 3, 4, etc

Humans have been using numbers to count with for thousands of years. It is a very natural thing to do.

You can have "3 friends", a field can have "6 cows" and so on.

So we have:

Counting Numbers: {1, 2, 3, ...}

And the "Counting Numbers" satisfied people for a long time.

ZeroThe idea of zero, though natural to us now, was not natural to early humans ... if there is nothing to count, how can you count it?

Placeholder

But about 3,000 years ago, when people started writing bigger numbers like "42" they had a problem: how to tell the difference between "4" and "40" ? Without the zero they look the same!

So they used a "placeholder", a space or special symbol, to show "there are no digits here"

5 2 So "5 2" meant "502"

(5 hundreds, nothing for the tens, and 2 units)

The idea of zero had begun, but it wasn't for another thousand years or so that people started thinking of it as an actual number.

But now we can think

"I had 3 oranges, then I ate the 3 oranges, now I have zero oranges...!"

The Whole NumbersSo, let us add zero to the counting numbers to make a new set of numbers.

But we need a new name, and that name is "Whole Numbers":

Whole Numbers: {0, 1, 2, 3, ...}

Negative NumbersBut the history of mathematics is all about people asking questions, and seeking the answers!

One of the good questions to ask is

"if you can go one way, can you go the opposite way?"

We can count forwards: 1, 2, 3, 4, ...

... but what if we count backwards:

3, 2, 1, 0, ... what happens next?

The answer is: you get negative numbers:

Now we can go forwards and backwards as far as we want

But how can a number be "negative"?

By simply being less than zero.

A simple example is temperature.

We define zero degrees Celsius (0° C) to be when water freezes ... but if we get colder than that we need negative temperatures.

So -20° C is 20° below Zero.

So negative numbers exist, and we're going to need a new set of numbers to include them ...

IntegersIf we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers

Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}

The Integers include zero, the counting numbers, and the negative of the counting numbers, to make a list of numbers that stretch in either direction infinitely.

 

FractionsYou took a number (1) and divided by another number (2) to come up with half

(1/2)

The same thing would have happened if you had four biscuits (4) and needed to share them among three people (3) ... they would get (4/3) biscuits each.

A new type of number, and a new name:

Rational NumbersAny number that can be written as a fraction is called a Rational Number.

So, if "p" and "q" are integers (remember we talked about integers), then p/q is a rational number.

Example: If p is 3 and q is 2, then p/q = 3/2 = 1.5 is a rational number

The only time this doesn't work is if q is zero, because dividing by zero is undefined.

Rational Numbers: {p/q : p and q are integers, q is not zero}

So half (½) is a rational number.

And 2 is a rational number also, because you could write it as 2/1

So, Rational Numbers include:

all the integers and all fractions.

Even a number like 13.3168980325 is a Rational Number.

That would seem to include all possible numbers, right?

But There Is MorePeople didn't stop asking the questions ...and here is one that caused a lot of fuss during the time of Pythagoras:

If you draw a square (of size "1"), what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

... in fact you cannot answer that question using a ratio of two numbers

... and so it is not a rational number

Wow! There are numbers that are NOT rational numbers! What do we call them?

What is "Not Rational"? Irrational!

Irrational Numbers

So, the square root of 2 (√2) is an irrational number. It is called irrational because it is not rational (can't be made using a simple ratio). It isn't crazy or anything, just not rational.

And we know there are many more irrational numbers. Pi (π) is a famous one.

Useful

So irrational numbers are useful. You need them to

find the diagonal distance across some squares, to work out lots of calculations with circles (using π), and more,

So we really should include them.

And so, we introduce a new set of numbers ...

Real NumbersThat's right, another name!

Real Numbers include:

the rational numbers, and the irrational numbers

Real Numbers: {x : x is a rational or an irrational number}

Any point Anywhere on the number line, that is surely enough numbers!

But there is one more number which has turned out to be very useful. And once again, it came from a question.

Imagine ...The question is:

"is there a square root of minus one?"

In other words, what can you multiply by itself to get -1?

Think about this: if you multiply any number by itself you can't get a negative result:

1×1 = 1, and also (-1)×(-1) = 1 (because a negative times a negative gives a positive)

So what number, when multiplied by itself, would result in -1?

This would normally not be possible, but ...

"if you can imagine it, then you can play with".

So, ...

Imaginary Numbers

... let us just imagine that the square root of minus one exists.

We can even give it a special symbol: the letter i

And we can use it to answer questions:

Example: what is the square root of -9 ?

Answer: √(-9) = √(9 × -1) = √(9) × √(-1) = 3 × √(-1) = 3i

OK, the answer still involves i, but it still gives a sensible and consistent answer.

And i has this interesting property that if you square it (i×i) you get -1 which is back to being a Real Number. In fact that is the correct definition:

Imaginary Number: A number whose square is a negative Real Number.

And i (the square root of -1) times any Real Number is an Imaginary Number. So these are all Imaginary Numbers:

3i -6i 0.05i πi

There are also many applications for Imaginary Numbers (for example in the fields of electricity and electronics), but those details are beyond this page.

Real and Imaginary NumbersImaginary Numbers were originally laughed at, and so got the name "imaginary". And Real Numbers got their name to distinguish them from the Imaginary Numbers.

But:

"what if you put a Real Number and an Imaginary Number together?"

Complex NumbersYes, if you put a Real Number and an Imaginary Number together you get a new type of number called a Complex Number here are some examples:

3 + 2i 27.2 - 11.05i

and a Real Number is also a Complex Number (with an imaginary part of 0):

4 (because it is 4 + 0i)

and likewise an Imaginary Number is also Complex Number (with a real part of 0):

7i (because it is 0 + 7i)

So the Complex Numbers include all Real Numbers and all Imaginary Numbers, and all combinations of them.

 

SummaryHere they are again:

Type of Number Quick DescriptionCounting or Natural Numbers {1, 2, 3, ...}Whole Numbers {0, 1, 2, 3, ...}Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}Rational Numbers p/q : p and q are integers, q is not zeroIrrational Numbers Not RationalReal Numbers Rationales and IrrationalsImaginary Numbers Squaring them gives a negative Real NumberComplex Numbers Combinations of Real and Imaginary Numbers

Common Number Sets

There are sets of numbers that are used so often that they have special names and symbols:

Symbol

Description

Natural Numbers The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics).

The set is {1,2,3,...} or {0,1,2,3,...}

Integers The whole numbers, {1,2,3,...} negative whole numbers {..., -3,-2,-1} and zero {0}. So the set is {..., -3, -2, -1, 0, 1, 2, 3, ...}

(Z is for the German "Zahlen", meaning numbers, because I is used for the set of imaginary numbers).

Rational Numbers The numbers you can make by dividing one integer by another (but not dividing by zero). In other words fractions.

Q is for "quotient" (because R is used for the set of real numbers).

Examples: 3/2 (=1.5), 8/4 (=2), 136/100 (=1.36), -1/1000 (=-0.001), etc.

Algebraic NumbersAny number that is a solution to a polynomial equations with rational coefficients.

Includes all Rational Numbers, and some Irrational Numbers.

Real NumbersAll rational and irrational numbers. They can also be positive, negative or zero.

Includes the Algebraic Numbers and Transcendental Numbers.

A simple way to think about the Real Numbers is: any point anywhere on the number line (not just the whole numbers).

Examples: 1.5, -12.3, 99, √2, π

They are called "Real" numbers because they are not Imaginary Numbers.

Imaginary NumbersNumbers that when squared give a negative result.

If you square a real number you always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 also, so "imaginary" numbers can seem impossible, but they are still useful!

Examples: √(-9) (=3i), 6i, -5.2i

The "unit" imaginary numbers is √(-1) (the square root of minus one), and its symbol is i, or sometimes j.

i2 = -1

Complex NumbersA combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary.

The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers.

Examples: 1 + i, 2 - 6i, -5.2i, 4

Illustration Natural numbers are a subset of Integers

Integers are a subset of Rational Numbers

Rational Numbers are a subset of the Real Numbers

Real Numbers together with Imaginary Numbers make up the Complex Numbers. 

3.5 Intervals

Interval on the real number line:Notation Interval Type Inequality Graph

[a,b] closed a ≤x≤ b [ ](a,b) open a <x< b ( )[a,b) half closed, half open a ≤x< b [ )(a,b] half open, half closed a <x≤ b ( ]

Notation Interval Type Inequality Graph [a,∞) half closed a ≥x [ (a, ∞) open a >x ( (-∞,b] half closed x ≤ b ](-∞,b) open x < b ( )(-∞,∞) closed -∞ <x< ∞

Example: Using inequalities to represent intervals of the following.a. c is at most 2b. m is at lest -3d. All x in the interval ( -3,5]

Solution: a. c ≤2b. m ≥-3d. -3<x ≤ 5

3.6 Real axis

The real axis is the line in the complex plane corresponding to zero imaginary part, Image [z]=0 . Every real number corresponds to a unique point on the real axis.

3.7 Exercises3.8 Reference