INTRODUCTION to SET THEORY - DISCRETE MATHEMATICS

Summary

This video script offers an introduction to set theory in discrete mathematics. It explains what sets are, how they are represented, and their properties such as finite or infinite nature, absence of order, and uniqueness of elements. The script also covers cardinality, set notation, common sets like natural numbers, integers, and rational numbers, and introduces set builder notation. Examples are provided to illustrate these concepts, making the foundational theory accessible.

Takeaways

• 🔢 Sets are collections of objects, known as elements, which can be represented visually or in curly braces.
• 🌀 Sets can be finite, like the numbers 1, 2, and 3, or infinite, like the set of all positive integers.
• 🔄 Elements in a set are unique, meaning repeated elements are only listed once, and the order of elements does not matter.
• 🌐 Common sets include natural numbers (starting from 0 or 1), integers (positive and negative whole numbers), and rational numbers (numbers that can be expressed as fractions).
• 📏 The size or cardinality of a set is the number of unique elements it contains. For example, the set {1, 2, 3} has a cardinality of 3.
• 0️⃣ The empty set, denoted as {}, has a cardinality of zero as it contains no elements.
• 🔍 The set containing only the empty set, {{}}, has a cardinality of 1 because it has one element: the empty set itself.
• ✏️ Set builder notation allows for a more formal representation of sets using rules and conditions, such as {x ∈ Z | x < 6} for all positive integers less than 6.
• 📦 Sets can contain other sets as elements. For example, a set containing an empty set and another set has two elements.
• 🧠 Understanding the difference between a set containing elements and a set containing other sets is crucial for grasping concepts like cardinality and set relations.

Q & A

What is a set in the context of set theory?

A set is a collection of distinct objects, which are called elements. It can be finite or infinite, and the order of elements does not matter.

How are sets typically represented visually?

Sets are often represented visually by drawing a circle around the elements, or using a Venn diagram for comparisons.

What is the formal way to write a set?

The formal way to write a set is by using curly braces to enclose the elements, listing them without repeating any element.

Can you provide an example of a finite set?

An example of a finite set is the set containing the numbers 1, 2, and 3, which can be written as {1, 2, 3}.

What is an infinite set and can you give an example?

An infinite set is a set with an unlimited number of elements. An example is the set of positive integers starting from 1 and going up to infinity.

Why are repeated elements in a set only listed once?

In set theory, repeated elements are listed only once to ensure that each element is unique within the set and to maintain the property of distinctness.

Is there an order to the elements in a set?

No, there is no order to the elements in a set. The set {3, 1, 2} is the same as {1, 2, 3}.

What are some common sets in mathematics?

Some common sets include the natural numbers (N), the integers (Z), the positive integers (Z+), and the rational numbers (Q).

How do you denote that an element is part of a set?

You use the element symbol followed by the set symbol (∈) to denote that an element is part of a set, such as 'a ∈ A'.

How is the size of a set represented?

The size of a set is represented by placing the absolute value bars around the set, like |C|, which denotes the cardinality of set C.

What is the empty set and how is it represented?

The empty set is a set that contains no elements and is represented by the symbol Ø or by curly braces {} in set notation.

What is the cardinality of the set containing the empty set?

The cardinality of the set containing the empty set is one, because the empty set itself is an element of the larger set.

What is set builder notation and how is it used?

Set builder notation is a way to define a set by specifying a property that elements must satisfy. It is used to describe sets with a large or infinite number of elements in a concise way.

Can you provide an example of set builder notation for rational numbers?

An example of set builder notation for rational numbers is {m/n | m, n are integers and n ≠ 0}, which includes all fractions where m and n are integers and n is not zero.

How do you determine the cardinality of a set given in set builder notation?

To determine the cardinality of a set in set builder notation, you would count the number of elements that satisfy the given condition.

Outlines

00:00

📘 Introduction to Sets and Set Theory

This paragraph introduces the concept of sets in mathematics. A set is defined as a collection of objects called elements. The concept of sets is explained using visual representations (like circles containing numbers), and formal notation using curly braces. Sets can be either finite or infinite, and key properties like no repetition of elements and no particular order in sets are discussed. Examples are given, such as finite sets (numbers 1, 2, 3) and infinite sets (positive integers). The distinction between visual representations and formal set notation is clarified.

05:01

🔢 Elements, Cardinality, and Empty Sets

This paragraph expands on the idea of set elements and how we denote them. It explains how elements belong to sets using the Epsilon symbol for membership and how to show when an element is not in a set. The concept of cardinality (the number of elements in a set) is introduced, with examples like a set of colors and the empty set. The difference between an empty set and a set containing an empty set is discussed, emphasizing that the set with an empty set has a cardinality of 1, while the empty set itself has a cardinality of 0.

10:01

🔠 Set Builder Notation and Predicate Notation

This paragraph introduces set builder notation, a method for defining sets more formally. An example is given for rational numbers, where elements are represented in the form of M over N, with conditions placed on M and N (both must be integers and N cannot be zero). Set builder notation is also applied to even integers, showing how every integer can be used to generate an even number in the set. A real-world example using objects on a desk is provided to further clarify the concept.

15:03

🧮 Exercises on Cardinality and Elements of Sets

This paragraph presents exercises on set theory, focusing on listing elements and determining the cardinality of sets. One exercise involves positive integers less than 6, and the cardinality of the resulting set is 5. Another exercise highlights the distinction between a set containing the empty set and a set containing other sets, reinforcing the visual and conceptual differences between these types of sets. The number of elements is determined by how many distinct 'boxes' (sets) are inside the set, ignoring the contents of those boxes.

Keywords

💡Set

A set is a fundamental concept in mathematics defined as a collection of distinct objects, considered as an object in its own right. In the video, sets are introduced as a way to group elements, which can be anything from numbers to words. The script uses visual representations like circles to depict sets and introduces notation such as curly braces to list elements within a set, exemplified by the set containing numbers 1, 2, and 3.

💡Elements

Elements are the members or objects that constitute a set. The video script explains that elements within a set are distinct and each element is listed without repetition, regardless of how many times it appears. For instance, the set containing 'a', 'b', 'a', 'c', 'b', 'a' is represented as {a, b, c}, highlighting that repeated elements are counted only once.

💡Finite and Infinite Sets

Finite sets contain a limited number of elements, while infinite sets have an unlimited number. The video uses examples such as the set of numbers between 1 and 9 (finite) and the set of positive integers extending to infinity (infinite) to illustrate these concepts. The dots in the script are used to denote the ongoing nature of infinite sets.

💡Order

Order refers to the sequence in which elements are arranged. The video clarifies that sets are unordered collections, meaning the sequence of elements does not matter. For example, the sets {3, 1}, {1, 2, 3}, and {2, 1, 3} are considered identical because sets do not consider the order of elements.

💡Cardinality

Cardinality is the number of elements in a set. The script explains how to determine the size of a set by counting its unique elements. For example, a set containing the primary colors yellow, blue, and red has a cardinality of 3, indicating there are three distinct elements.

💡Natural Numbers

Natural numbers are used to count the number of elements in a set. The video discusses a common debate about whether natural numbers start with 0 or 1. The script uses the terms 'positive integers' for natural numbers starting with 1 and 'natural numbers' for those starting with 0, emphasizing the importance of specifying the starting point when discussing these sets.

💡Integers

Integers include all whole numbers, both positive and negative, including zero. The video uses the set notation Z to represent all integers, which extends from negative infinity to positive infinity. This concept is crucial for understanding discrete mathematics, where integers are often used to represent quantities or positions.

💡Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers. The script initially lists examples like 1/1, 1/2, 1/3, etc., and later introduces set builder notation to define rational numbers more formally, showing how any fraction where the denominator is not zero can be part of the set of rational numbers.

💡Empty Set

The empty set, denoted by curly braces with nothing between them, is a set with no elements. The video explains that the size or cardinality of the empty set is zero. It also introduces the concept of a set containing the empty set, which has a cardinality of one, as it contains one element—the empty set itself.

💡Set Builder Notation

Set builder notation is a way to define a set by specifying a rule or condition that its elements must satisfy. The video uses examples such as defining even integers as {2n | n is an integer}, which means the set contains all numbers that are twice an integer. This notation is a powerful tool for describing sets without listing every element.

Highlights

Introduction to set theory as a fundamental notion in mathematics.

Definition of a set as a collection of objects called elements.

Visual representation of sets using circles.

Formal representation of sets using curly braces notation.

Sets can be finite or infinite.

Infinite sets represented with dots indicating a pattern that continues indefinitely.

Sets do not list repeated elements more than once.

Order is not significant in sets.

Introduction to common sets such as natural numbers and integers.

Explanation of the difference between natural numbers starting with 0 and positive integers starting with 1.

Definition of integers and rational numbers.

Introduction to the concept of elements and cardinality in sets.

Notation for expressing that an element is part of a set.

Notation for expressing that an element is not part of a set.

Explanation of the cardinality of a set and how to denote it.

Introduction to the empty set and its cardinality.

Discussion on the size of a set containing the empty set.

Introduction to set builder notation as a way to define sets.

Example of set builder notation for rational numbers.

Example of set builder notation for even integers.

Linguistic example of set builder notation using items on a desk.

Exercises to practice understanding of set theory concepts.

Explanation of cardinality for sets containing other sets.

Summary and conclusion of the set theory introduction video.

Transcripts

welcome to discrete mathematics let's

start with set theory which is a very

fundamental notion in the entirety of

mathematics so what is a set a set is a

collection of objects called elements

and this is very vague because we can

have a set about anything we want and

usually we talk about sets by drawing

little circles so for instance if I want

to set up let's say the numbers 1 2 & 3

I can draw a circle I can put the

numbers 1 2 & 3 in there and that is a

set containing numbers 1 2 & 3 and we

can write this formally in curly braces

and writing all of the elements inside

so in this case our set contains 1 2 & 3

and we can give this a label and maybe

call this a so these are all different

ways of representing the same thing a is

the set 1 2 & 3 which looks like a

circle containing elements 1 2 & 3 of

course this is just visual and this

notation in curly braces is called a

lists notation because you're listing

them all now sets can be finite or they

can be infinite so we can have a set a

containing all the numbers between 1 & 9

or we can have the set of positive

integers that go from 1 all the way up

to infinity and in this case this is an

infinite set

so these dots mean that there's an

implied pattern that just goes on

forever so here that plus if we have 1 2

3 4 we put dot dot we mean that it goes

5 6 7 8 9 10 so on and so forth so those

are really the fundamental notions of

sets now there's some additional points

these sets and that is that repeated

elements are only listed once so if we

have a set a bacb

a even though a is repeated three times

we only write at once B is repeated

twice we only write at once and C is

repeated once so this set a BA CBA is

the same set as ABC and they would both

be drawn as circles with some element a

some element B and some elements C if we

don't care about the amount of numbers

of these things in there we don't care

if there's three A's we don't care if

there's two B's we just want to know

what's inside our circle there's also no

order in a set and then a diagram this

is easy to see because there's no order

in a circular diagram but essentially

the set three to one is the same thing

as a set one two three and it's also the

same thing as the set to one three so on

and so forth there's no order to these

sets so those are two pretty important

parts usually on an exam you might be

given a set that has repeated elements

and they might ask you how big is this

set how many things are in it and if

you're not sure about repeated elements

then you might get the wrong answer

there's a few common sets that we should

talk about one of them is the natural

numbers the first time I made this video

controversy and there's still some

controversy but there's two ways to do

the natural numbers one of them is to

start with zero do 1 2 3 all the way up

forever and ever a another group of

mathematicians starts the natural

numbers with 1 2 3 so on and so forth

I'm going to refer to the natural

numbers starting with 1 as the positive

integers and I talk about natural

numbers anywhere else I will usually

mean including 0 but I may be more

specific about it so I will specify

whether it includes 0 or not but

typically with 1 2 3 so on and so forth

I'll use Z plus the integers are a set

of numbers which are either positive or

negative and they're whole numbers so

all the way from negative infinity

negative 2 negative 1 0 1 2 all the way

to positive infinity so that is Z that's

very important notion which we

use a lot in discrete math and then

rational numbers all these kind of hard

to list so I'm going to cheat a little

bit for now and I'm going to say well we

have like 1 over 1 1 over 2 1 over 3 2

over 3 so on and so forth and in a few

slides we'll figure out a better way to

write this

so of course rational numbers are any

numbers you can write as a fraction so

elements and cardinality now we know

what sets are now we know some

additional pointers about sets we know

some common sets but we want to be able

to talk about the things in those sets

and how big those sets are so I have C

and I have a set containing yellow blue

and red so these are the primary colors

fact sets don't have to be all

mathematics they can be words too and

have meaning in the real world so I want

to say something like yellow is an

element of C and this means that yellow

is inside of our set C now how do we

write this well we have an element

yellow and we want to say it's part of a

set C so in order to write this in

notation we use this Epsilon so yellow

is in the set C what about saying green

is not in the set C so I want to say

Green is not in here in fact we don't

see it in the set C so what were you to

say Green is not in C well we use the

set membership symbol and we just put a

line through it to mean Green is not in

our set see how do we talk about the

size of this I would say the size of C

is 3 which means the amount of things in

C is 3 there are 3 different elements in

C well we just draw these absolute value

bars around it so we say that the

absolute value or the size of C is equal

to 3 and that's just that the

cardinality of C is 3 so now we can talk

about whether things are intercepts and

how many things are in our sets

now there's one particularly interesting

set that contains nothing at this is

called the empty set and this is written

with this symbol and if I wanted to

write it in the set notation with curly

braces it would look like this it has a

left curly brace or right curly brace

and there's nothing inside of it because

it's empty so what's the size of this

set well there's nothing in it so the

size of it is zero and there's only one

set that has a size of zero and that is

the empty set so here's the question and

I'm throwing this right at you right

away because this trips people up all

the time what is the size of the set

containing the empty set

so I'm asking if I have a set containing

the empty set what's the size of that

set

well this empty set is an element of the

larger set so the size of the set

containing the empty set is one because

this set itself there's a big set has

one element it has an empty set as an

element so sets can have sets as

elements we'll see another example later

but typically I like to look where the

commas are so if they like a big set and

a comma then another big set and that's

all it's in the set and there be two

elements in this case we have these

empty braces but there's something

inside of it so therefore there's an

element in that set this is difference

of course than the size of the set

containing nothing this would be zero so

we can see the contrast between these

two sets this is the empty set on the

right well this is a set containing the

empty set on the left the left one has a

size of 1 the right one has a size of

zero and we can see the differences

visually okay so that's the empty set

there'll be more questions about the

empty set when we get to the subsets

video so more tricky stuff will happen

there

the next thing we should talk about is

set builder notation so before I

introduce the rational numbers MSI Delta

is 1 over 1 1 over 2 1 over 3 2 over 3

so on and so forth

it's going to be a better way of

representing this entire set and this is

set builder notation or we can call it

maybe predicate notation where we define

elements as variables so for instance I

can say this is the set containing

elements that are of the form M over m

such that so the straight bar means such

that and now I'm going to give a rule so

the left is like the form that it takes

and on the right side we're going to

give it a rule we're going to say that m

and n have to be integers and also M has

to not be 0 so that way we're not

dividing by 0 and this expresses the

entire set of rational numbers I'm

saying look take any two integers m and

n we could put em on top of N and as

long as n is in 0 it's going to be in

that set soul do another example with

even integers so we can write all the

way to negative infinity negative 4

negative 2 0 2 & 4 so on and so forth to

infinity or we could do this in set

builder notation so we could say this is

the set of 2 n such that n is an integer

so what this means is we take an integer

n so let's say we take one then in our

set we add 2 times that integer so 2

times 1 so we add 2 let's say we take 2

then we take 2 times 2 and put it are

set for let's say we take negative 1

when we take negative 1 times 2 and then

we get negative 2 in our set as well and

this just goes on forever so for every n

that is an integer we add 2n to our set

and that gives us a set of even integers

so here's a more linguistic example so

something we can

flying in the real world to see how this

notation works I have a desk and on this

desk

I have a drink I have a laptop and I

have a microphone and there's some other

things on my desk that I could list but

let's just say my desk only has these

three things now this is one way I can

talk about things I can say I got these

two strengths of soft office microphone

on my desk or I could write this in set

builder notation if I might have a lot

more things or maybe I want to be bagged

or maybe I'm not entirely sure what's on

my desk but I know there's things on my

desk so I could rewrite this as the

scent of X so this is a variable X and

what's the condition for X X is on my

desk yeah so this is kind of cheating

right but they're the same set so the

first one I'm listing all these items

individually the second one I'm saying

look it's the set of variables X and

whatever X is it just happens to be on

my desk so everything that's on my desk

is just going to shove it in as X and

it's going to build a set with it so

that's the set builder notation and of

course mathematically we do this very

formally with the course numbers and

formulas and notation while of course if

you do a linguistics example with words

it's much more wordy it's much more

flexible much more floaty

but they both convey the same meaning so

if you didn't quite understand the

mathematic example then hopefully you'll

be able to translate this desk example

back into the mathematics after all

maybe understanding this at a more

fundamental level so here I have some

exercises and these all refer to the or

the first two refer to the set D as the

third one is completely different try

them yourself and if not well I'll give

you the answers now so list the elements

of D D is the set of X in positive

integers such that X is less than 6 so

I'm going to take all X less than 6

and adds into the set if it's positive

integer so this means that I can have

one because one is a positive integer n

is less than six let's do this in a

different color two three four and five

can I add six I can't add six because X

is not less than six can I add zero well

I can't add zero because I want the X's

that are in positive integers and zero

is not a positive integer

okay what's the cardinality of D then

well this is easier because I've already

written out all the elements in my set

so I see there's five things there so I

can say the cardinality of B is equal to

five because it contains the numbers one

two three four and five

now here's another question and this is

what I alluded to before what is the

cardinality of the set containing the

empty set and the set a B okay so how

many elements are in the set and when we

look at sense it's kind of like looking

into a box and saying well what can I

see I see an empty set and I see another

set so really what I see inside this box

if I were to draw this out is a box

containing nothing and a box containing

a and B so the stuff inside of these

boxes essentially with a cardinality

question these things are invisible you

can't see what's going on so the

question is how many things do we see

inside that set what is the size how

many elements are in that set that big

box and the answer to this is two and so

I said look at the commas here so we

have an empty set which is the first

thing then we have a comma then we have

this other entire set so there's two

things we can see now how does this

differ from let's say if we have a set

containing the empty set the set

a and then the set containing B well

this is different

of course so in this example if I were

to draw this we have a box containing

nothing in it we have a box with a in it

and then we have another box with being

it so how many elements do we see well

we can't see inside these boxes but if

we just open our box and take a look

we'll see three other boxes so in other

words the size of this would be three

one more example let's say we took away

one of these so now I have something

that looks like this I have the empty

set I have the element a and I have a

set containing B once again there are

three elements that we can see we can

see this box containing something they

can see the element a and then we can

see this box containing something else

which happens to be B so once again this

cardinality is equal to three so that is

it for the introduction to set Theory

video if you have any questions please

leave them in the comments below and I

will do my best to answer them