Discrete Math - 2.1.1 Introduction to Sets

Summary

This educational video offers an introduction to the concept of sets in mathematics, covering essential vocabulary and notation. It explains that a set is an unordered collection of distinct objects, using the example of siblings to illustrate. The video introduces various types of sets, including natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), real numbers (R), and complex numbers (C). It also discusses different notations for representing sets, such as roster notation for discrete sets and set builder notation for both discrete and continuous sets. Interval notation for continuous sets is also explained, along with the concepts of the universal set and the empty set, setting the stage for further exploration of set relationships.

Takeaways

• 📚 A set is an unordered collection of distinct objects.
• 👦 Elements of a set are separated by commas and can be listed in any order.
• 📝 The notation '∈' means 'is an element of' or 'is contained within' a set.
• 🔢 Sets have specific notations like 'N' for natural numbers, 'W' for whole numbers, 'Z' for integers, 'Q' for rational numbers, 'R' for real numbers, and 'C' for complex numbers.
• 📈 Roster notation lists the elements of a set and is used for discrete sets.
• 📐 Set builder notation uses a descriptive rule to define the set and can be used for both discrete and continuous sets.
• 📊 Interval notation is used for continuous sets and specifies the range of values using brackets.
• 🌐 The universal set contains all elements under consideration in a given context.
• ⛔ The empty set, denoted by 'Ø' or '{}', contains no elements.
• 🔗 Set relationships, such as subsets and supersets, will be explored in further detail.

Q & A

What is a set in the context of this video?

A set is an unordered collection of objects, where the order of elements does not matter.

What does the notation '∈' represent in set theory?

The notation '∈' represents 'is an element of' or 'is contained within' a set.

What are the symbols for natural numbers, whole numbers, integers, rational numbers, real numbers, and complex numbers?

The symbols are N for natural numbers, W for whole numbers, Z for integers, Q for rational numbers, R for real numbers, and C for complex numbers.

What is the difference between a discrete set and a continuous set?

A discrete set is countable and consists of distinct, separate elements, while a continuous set has an infinite number of values within a range and is not countable.

Why can't roster notation be used for continuous sets?

Roster notation is not suitable for continuous sets because it requires listing all elements, which is impossible for infinite sets like continuous intervals.

What is set builder notation and when is it used?

Set builder notation is a way to describe a set by stating a condition that its elements must satisfy. It can be used for both discrete and continuous sets.

How do you represent an interval in interval notation?

Interval notation is represented by listing the endpoints of the interval and indicating whether they are included or not using brackets. For example, [0, 1] includes both 0 and 1.

What is a universal set?

A universal set is the set of all elements under consideration in a particular context or problem.

How is the empty set represented and what does it signify?

The empty set is represented by a slash through the number 0 (∅) or by set brackets with nothing inside ({}). It signifies a set with no elements.

What are the two special sets mentioned in the video?

The two special sets mentioned are the universal set, which contains all elements under consideration, and the empty set, which contains no elements.

Outlines

00:00

📚 Introduction to Sets

This paragraph introduces the concept of sets, emphasizing that a set is an unordered collection of objects. It uses the example of siblings to illustrate that elements in a set are not ordered. The paragraph explains the notation for set membership and non-membership, and introduces various types of sets, including natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), real numbers (R), and complex numbers (C). It also mentions special notations for positive and negative integers. The paragraph concludes by discussing different types of set notation, such as roster notation, which is used for listing elements in discrete sets.

05:04

📐 Set Notation and Discrete vs. Continuous Sets

The second paragraph delves into the difference between discrete and continuous sets, with discrete sets being countable and continuous sets having an infinite number of values within a range. Set builder notation is introduced as a flexible method for describing both types of sets. The paragraph provides examples of how to use set builder notation to define sets with specific properties, such as being natural numbers less than or equal to 5. It then transitions into interval notation, which is particularly suited for continuous sets. The paragraph explains how to use interval notation to represent sets on a number line, including the use of closed and open brackets to indicate whether endpoints are included in the set.

10:05

🔍 Special Sets and Set Relationships

The final paragraph discusses special sets such as the universal set, which contains all elements under consideration, and the empty set, which contains no elements. It highlights the importance of not confusing the empty set with a set containing the empty set as an element. The paragraph sets the stage for exploring set relationships in subsequent content, suggesting that upcoming discussions will cover topics like subsets and other relationships between sets.

Keywords

💡Set

A set is defined as an unordered collection of distinct objects, which can be numbers, people, or any other items. In the video, sets are introduced as fundamental to understanding various mathematical operations and concepts. The script uses the example of the speaker's brothers to illustrate a set, emphasizing the unordered nature of sets, where the elements 'Eric, Adam, Kevin' are listed without any specific order.

💡Element

An element refers to a member or an object within a set. The video clarifies that elements are the constituents of a set, and it uses the notation '∈' to denote that an object is an element of a set. For instance, the script mentions that 'Adam' is an element of set B, while 'Larry' is not, highlighting the distinction between included and non-included elements.

💡Natural Numbers

Natural numbers are the set of positive integers starting from 1 and extending to infinity. The video script introduces this set as the basic counting numbers, which are fundamental in mathematics. The script uses the symbol 'N' to represent natural numbers, emphasizing their role in counting and arithmetic operations.

💡Whole Numbers

Whole numbers include all natural numbers as well as zero. The video script uses the symbol 'W' to denote whole numbers, which are crucial for understanding integers and their properties. The script explains that whole numbers are the same as natural numbers but start with zero, thus including 0, 1, 2, and so on.

💡Integers

Integers are a set of whole numbers that also include negative counterparts and zero. The video uses the symbol 'Z' to represent integers, which are essential for understanding operations involving positive and negative values. The script describes integers as extending from negative infinity to positive infinity, covering all whole numbers and their negatives.

💡Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. The video script uses 'Q' to denote rational numbers, highlighting their importance in algebra and number theory. The script explains that rational numbers include fractions and integers, as they can be expressed in the form of a/b, where a and b are integers and b ≠ 0.

💡Real Numbers

Real numbers encompass all the numbers on the number line, including integers, rational numbers, and irrational numbers. The video script uses 'R' to represent real numbers, which form the basis for continuous mathematical analysis. The script clarifies that real numbers include all the numbers that are not imaginary, thus covering a vast range of mathematical applications.

💡Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, represented in the form a + bi. The video script uses 'C' to denote complex numbers, which are crucial for understanding advanced mathematical concepts. The script explains that complex numbers extend the number system to include solutions to equations that have no real solutions.

💡Roster Notation

Roster notation is a method of representing a set by listing its elements within curly brackets. The video script explains that this notation is straightforward and is used for discrete sets, where elements can be counted and listed. The script provides an example of roster notation with the set {1, 2, 3, 4, 5}, illustrating how elements are simply listed.

💡Set Builder Notation

Set builder notation is a way to describe a set by specifying a property that all elements of the set share. The video script uses this notation to represent sets where elements are defined by a condition, such as 'X is an element of the natural numbers and X is less than or equal to 5'. The script highlights that set builder notation is versatile and can be used for both discrete and continuous sets.

💡Interval Notation

Interval notation is used to represent sets of real numbers that include all numbers within a specific range. The video script explains interval notation by using examples like [0, 1], which includes all numbers from 0 to 1, inclusive. The script clarifies that this notation is particularly useful for continuous sets and helps in visualizing the set on a number line.

Highlights

A set is an unordered collection of objects.

Elements of a set are separated by commas and listed within curly braces.

The symbol '∈' denotes that an element is contained within a set.

Natural numbers are represented by the set N and include counting numbers starting from 1.

Whole numbers include zero and are represented by the set W.

Integers, denoted by Z, are positive and negative whole numbers without negative zero.

Rational numbers, represented by Q, are quotients of two integers where the denominator is not zero.

Real numbers, denoted by R, include all natural, whole, and rational numbers, and exclude imaginary numbers.

Complex numbers, represented by C, are in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.

Positive integers are denoted by Z+ and negative integers by Z-.

Roster notation lists elements of a set and is used for discrete sets.

Set builder notation describes the elements of a set using a condition and can be used for both discrete and continuous sets.

Interval notation is used for continuous sets and indicates the range of values included in the set.

Closed brackets in interval notation indicate that the endpoint is included in the set.

Open brackets in interval notation indicate that the endpoint is not included in the set.

The universal set contains all elements under consideration in a particular study.

The empty set, denoted by ∅ or {} without contents, is a set with no elements.

Transcripts

this video is an introduction to sets

and that includes both vocabulary and

notation before we begin working with

sets it's important to understand what a

set is so a set is simply an unordered

collection of objects so for instance if

I wanted to tell you that I have three

brothers I could say the set of people

who are my brothers include Eric Adam

Kevin in no particular order so I didn't

write them youngest to oldest oldest to

youngest favorite to least favorite etc

just unordered an element is simply an

object in the set so the elements of set

B are Eric Adams Kevin and notice

they're just separated with a comma and

then of course we have the notation here

so this means is an element in the set

or is contained within the set so I

could say Adam belongs to set B Larry

does not belong to set B because I don't

have a brother named Larry so it means

is contained in or is an element in and

of course not contained or not an

element in there are several sets that

you should know and it's very possible

that you already know all of these but

I'm going to go through them anyway just

to be safe so the first that you should

know is the set of natural numbers the

set of natural numbers are just the

counting numbers so if I wanted to start

listing them it would be 1 2 3 on to

infinity W represents whole numbers and

whole numbers

our zero and the natural numbers which

means same set but starting with zero

instead then we have Z which represents

the integers and the integers are just

the positive and negative whole numbers

and of course there is no negative zero

but we get the idea that it would be on

to negative infinity negative 2 negative

1 0 1 2 onto positive infinity then we

have the rational numbers which you

would think would be an R but obviously

we have saved the R for something else

Q represents rational numbers because a

rational number is a quotient of the two

terms a and B where a and B are both

integers B is not 0 because of course we

cannot divide by zero and a over B is in

lowest terms then we have R which

represents our real numbers and the real

numbers is everything above it so all

the natural all the whole all the

integers all the rational those are all

real numbers so anything that is not

imaginary is real and that brings us to

C which represents our complex numbers

and complex numbers are numbers written

in the form a plus bi where a actually

is a real number it's a the real

component of a complex number but as you

can see B I that I would represent the

imaginary portion of that number so

that's all the sets you should know one

more thing I do want to point out

sometimes you're going to see something

like Z with a plus or Z with a minus and

that would just represent the positive

integers

or if you're negative it would be the

negative integers now let's talk about

different types of set notation so

there's roster notation that is very

straight forward roster notation just

like the roster of a sports team is

simply listing the elements in the set

so you can see set s I have said is 1 2

3 4 & 5 so roster notation would just be

listing those roster notation needs to

be used with discrete sets and what do I

mean by discrete sets well hopefully we

know since here we are in discrete math

studying discrete objects discrete sets

include things that are countable they

are not continuous so let me give you an

example of something that would be not

discrete would be 0 is less than or

equal to X is less than or equal to 1 if

I'm looking at all of the values between

0 & 1 there are quite a few there are

actually infinite number of values

between 0 & 1 because they could say

point 1.2.3.4 etc but then I could say

0.1 1.2 1.3 1 you get the idea I can

just keep adding another decimal place

and that gives me a whole nother you

know bunch of numbers that can be

included so this is a continuous set and

therefore I cannot use roster notation

with that set has to be a discrete set

then we have set builder notation now

set builder notation can be used for any

kind of set so it could be a continuous

set or it could be a discrete set so let

me show you an example still using set s

if I wanted to write set s in another

way I would say set s is all of the

elements X such that so this little line

here just means such that and everything

that follows the such that line

is describing the elements in the set so

in this case I could say all of the

elements in the set such that X is an

element of the natural numbers remember

the natural numbers with just the

counting numbers and also that X is less

than or equal to 5 does that describe my

set

well absolutely it does it describes my

set but also I could have said all of

the elements X such that X is an integer

well now integers include the negative

and positive values so here I would have

to say that 0

I'm sorry 1 is less than or equal to X

is less than or equal to 5 so that's

another way to correctly talk about set

s and set builder notation now let's

talk about interval notation so I'm

going to start with set builder notation

and we're going to move to interval

notation let's we have to have a

different set now because interval

notation is what we're going to do with

these continuous sets so we have to use

that interval notation or you can use

set builder but most often interval is

the best it's the clearest it's the

easiest so let's take a look let's say

set B is equal to all of the X's such

that 0 is less than or equal to X is

less than or equal to 1 so it's

including all of the values between 0

and 1 inclusive so 0 is included and 1

is included so if I'm looking at a

number line where I have 0 and 1 and

this might bring you back to middle

school days if I were to graph on a

number line all of the values that are

included in set B 0 would be included 1

would be included and all of the values

between 0 & 1 are also included now if I

want to rewrite this in interval

notation I'm going to pay special

attention to the endpoints of my

interval and

whether or not those endpoints are

included so if I'm looking at the end

points 0 & 1 are the end points so 0

comma 1 an interval notation again is

only for continuous sets so I don't have

to say that it's continuous interval

notation implies it's continuous now 0

is included in this set so I'm going to

use a what's called a closed bracket and

1 is also included in the set and again

because these both say or equal to and

so that's how I would write that in

interval notation 0 comma 1 with the

brackets on the outside which are closed

records now what if instead I said let's

let's set M represent all of the X's

such that 0 is less than X is less than

or equal to 1 so what did I change here

well I still have 0 and 1 but now notice

0 is not included because it's not 0 is

less than or equal to X which means 0 is

not included there we would use an open

bracket now I've still included the 1 so

the 1 would still be closed or of course

I could have said that 0 is less than X

is less than 1 and then I would have

open brackets on each side let's take a

look at two special sets the first is

not a term that you're going to hear

often but the concept is important so

the first term or concept is the

universal set and the universal set is

essentially the set of all elements

under consideration which essentially

just means what am i studying so let's

say for instance I was going to create a

Venn diagram and I'm interested only in

the natural numbers so this box would

represent the universal set of natural

numbers so that's how I would denote

that now hopefully you

with a Venn diagram before so anything

that goes in this box is a natural

number and let's say then for instance I

also had a subset which we're going to

talk about more in just a little bit

let's say this subset is all of the even

natural numbers so where were the number

one go well number one is a natural

number it's a counting number so it

would go within the universal set of

natural numbers but it wouldn't go in

here where I'm only containing the even

Naturals but I would put the number two

number three would go out here somewhere

number four would go inside number five

would be out here somewhere number six

would go inside you get the idea the box

itself is the universal set and that's

the set containing everything under

consideration and then of course we'll

talk more about some sense in a little

bit then there's the empty set and the

empty set is simply a set with no

elements and you'll see that denoted

sometimes as a zero with a line through

it like this or you might see the set

brackets but with nothing inside which

of course be the empty set what I do not

want to see you do is give me the set

brackets around the empty set because

that would imply that this is a set

containing one element and that one

element is the empty set so don't do

this coming up next we're going to

explore some set relationships