Sets & Set Operations (Introduction)

Summary

Houston Math Prep's video script introduces fundamental concepts of set theory, including defining sets and their elements. It explains notations like curly braces and set builder notation, and operations such as intersection, union, and complement. The script also covers subset relationships and the universal set, using days of the week as examples. It aims to clarify set operations and their symbolic representations, making the abstract concept of sets more tangible.

Takeaways

• 📚 A set is a well-defined collection of distinct objects, with no ambiguity about its elements.
• 🔤 Sets are commonly denoted by capital letters, and elements are represented within curly braces.
• 📈 Set builder notation is a convenient way to define sets, especially when they have many elements, by specifying a rule for membership.
• 🗓️ The script uses weekdays and weekends as examples to illustrate set A and set B, respectively.
• 🔶 The symbol for 'element of' is a rounded object, indicating that a specific item belongs to a set.
• ❌ The symbol for 'not an element of' is a rounded object with a slash through it, indicating exclusion from a set.
• 🤝 The intersection of two sets (∩) includes only the elements that are common to both sets.
• 🔄 The union of two sets (∪) includes all elements from both sets, whether they are unique or shared.
• 🈳 The empty set, denoted by a circle with a slash, represents a set with no elements, which can result from an intersection of disjoint sets.
• 📖 The complement of a set consists of all elements in the universal set that are not in the given set, denoted with an apostrophe next to the set symbol.
• 👉 A subset (⊆) is a set where all its elements are also found in another set, indicating a 'contained within' relationship.

Q & A

What is a set in the context of mathematics?

A set is a well-defined collection of distinct objects, where there is no ambiguity about what is included in the set and what is not.

What are the objects within a set called?

The objects within a set are called elements or members of the set.

Why are capital letters commonly used to denote sets?

Capital letters are commonly used to denote sets to easily refer to them without having to write out the entire description each time.

How are elements of a set represented in mathematical notation?

Elements of a set are represented within curly braces, separated by commas.

What is set builder notation and how is it used?

Set builder notation is a way to define a set by specifying a rule or condition for the elements, rather than listing them out individually. It is useful for sets with a large number of elements.

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

To denote that an element is part of a set, you use the 'element of' symbol (∈) followed by the set name, such as 'Thursday ∈ A'.

What does the symbol '∉' represent in set notation?

The symbol '∉' represents that an element is not a member of a set, indicating exclusion.

What is the intersection of two sets and how is it denoted?

The intersection of two sets is the set containing all elements that are common to both sets. It is denoted using the symbol '∩', such as 'C ∩ D'.

What is the union of two sets and how is it represented?

The union of two sets is the set containing all elements that are in either of the sets, or in both. It is represented using the symbol '∪', such as 'C ∪ D'.

What is the empty set and how is it notated?

The empty set is a set with no elements. It is notated with a circle containing a diagonal slash, such as '∅'.

What does it mean for two sets to be disjoint?

Two sets are considered disjoint if they have no elements in common, meaning their intersection is the empty set.

What is the complement of a set and how is it denoted?

The complement of a set includes all elements in the universal set that are not in the given set. It is denoted with a prime symbol next to the set name, such as 'C'.

What is a subset and how is it represented?

A subset is a set where all of its elements are also elements of another set. It is represented using the subset symbol '⊆', such as 'E ⊆ F'.

Why is the empty set considered a subset of any set?

The empty set is considered a subset of any set because it has no elements, and thus none of its elements are missing from any other set.

Outlines

00:00

📚 Introduction to Sets and Set Operations

This paragraph introduces the concept of sets in mathematics, emphasizing that a set is a well-defined collection of distinct objects without ambiguity. The elements of a set are denoted using capital letters, and sets are represented using curly braces with elements separated by commas. The paragraph explains two methods of representing sets: listing elements directly and using set builder notation, which is useful for large sets. It also covers the notation for elements belonging to a set and not belonging to a set. The concept of intersection and union of sets is introduced with examples, explaining that the intersection includes elements common to both sets, while the union includes all elements from both sets. The paragraph concludes with the concept of the empty set, which contains no elements, and the idea of disjoint sets, which have no elements in common.

05:00

🔄 Set Operations: Union, Complement, and Subsets

The second paragraph delves deeper into set operations, focusing on the union of sets, which includes all elements from at least one of the sets. It introduces the concept of a universal set, which contains all possible elements under consideration. The paragraph then explains the complement of a set, which consists of all elements in the universal set that are not in the given set. The concept of subsets is introduced, where one set is considered a subset of another if all its elements are also in the other set. The paragraph clarifies that the empty set is a subset of any set because it contains no elements that could be missing from another set. The video script concludes with a brief mention of the subset notation and the fact that the empty set is a subset of any set, emphasizing the foundational concepts of set theory and their practical applications in mathematics.

Keywords

💡Set

A set is a well-defined collection of distinct objects, where each object is an element or member of the set. In the context of the video, sets are used to represent collections of days such as weekdays and weekends. The video emphasizes that sets should be unambiguous, meaning there is no doubt as to what is included in the set. For example, set A represents weekdays, and set B represents weekends, making it clear what elements belong to each set.

💡Elements

Elements are the objects that belong to a set. The video explains that elements are what make up a set, and they are the building blocks for defining the set. For instance, the elements of set A are the weekdays Monday through Friday, and the elements of set B are the weekend days, Saturday and Sunday.

💡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 this notation to describe set A as the set of all things X such that X is a weekday, and set B as the set of all things X such that X is a weekend day. This notation is particularly useful when the set has many elements, as it avoids the need to list them all individually.

💡Intersection

The intersection of two sets is the set of elements that are common to both sets. In the video, the intersection of sets C and D, which represent days with classes, is calculated. The video demonstrates that Monday and Friday are the days common to both sets, hence they are the elements of the intersection of C and D. This operation is denoted by the symbol ∩.

💡Union

The union of two sets is the set of all elements that are in either set. The video explains the union of sets C and D by combining all the days listed in both sets, resulting in a list of all days from Monday to Saturday. The union operation includes elements that are in at least one of the sets and is denoted by the symbol ∪.

💡Empty Set

The empty set is a set with no elements. The video mentions the empty set when discussing the intersection of sets A and B, which have no common elements. The empty set is denoted by a circle with a diagonal slash through it. It represents the absence of any elements and is a subset of every set, including the universal set.

💡Universal Set

The universal set is a set that contains all the elements under consideration in a particular context. In the video, the universal set is all the days of the week, which is used as a reference when discussing the union of sets A and B. The union of A and B includes all the days, thus representing the universal set for the given scenario.

💡Complement

The complement of a set is the set of all elements in the universal set that are not in the given set. The video illustrates this by defining the complement of set C as the days of the week when there are no classes, which are Wednesday, Saturday, and Sunday. The complement is denoted by a symbol that looks like an apostrophe next to the set symbol.

💡Subset

A set E is a subset of another set F if every element of E is also an element of F. The video uses the subset symbol (⊆) to indicate this relationship. For example, set C, which contains days with classes, is a subset of set A, which contains all weekdays, because all elements of C are also in A.

💡Disjoint Sets

Disjoint sets are sets that have no elements in common. The video refers to sets A and B as disjoint because they represent weekdays and weekends, respectively, and there are no days that are both a weekday and a weekend. Disjoint sets highlight the concept of sets having no overlap.

Highlights

A set is a well-defined collection of distinct objects with no ambiguity about its elements.

Elements of a set are referred to as members.

Sets are commonly denoted by capital letters, starting with those early in the alphabet.

Curly braces are used to enclose the elements of a set, with elements separated by commas.

Set builder notation is an alternative way to define sets, especially when they have many elements.

The symbol '∈' is used to denote that an element is a member of a set.

The symbol '∉' with a slash indicates that an element is not a member of a set.

Intersection of sets (∩) represents elements common to two sets.

Union of sets (∪) includes elements that are in at least one of the sets.

An empty set (∅) is a set with no elements and is represented by a circle with a slash.

Disjoint sets are sets that have no elements in common.

The universal set contains all elements under consideration in a given context.

The complement of a set (C) includes all elements in the universal set that are not in the set.

A subset (⊆) is a set where all its elements are also found in another set.

The empty set is a subset of any set because it contains no elements that could be missing from another set.

Subset notation with a slash (⊈) indicates that one set is not a subset of another.

The video provides a comprehensive introduction to set theory and operations, using days of the week as examples.

Transcripts

hey everyone Houston math prep here we

want to introduce to you some

information about sets and set

operations so a set is a well defined

collection of objects that just means we

don't want there to be any ambiguity or

any doubt as to what is in our set and

what is not in our set the objects in

our set are called the elements of the

set or the members of the set they

belong to the set so here I've got two

sets I've got a set called a in a set

called B it's very common that we name

sets using capital letters usually

starting at the beginning of the

alphabet but we can really start

anywhere we want usually capital letters

though are used to denote sets so that

we can refer to them easily and not have

to write out an entire description to

refer to the set so if I just say set a

you know I'm talking about weekdays and

if I say set B you know I'm talking

about weekends without me having to

write that out or describe it fully you

can tell that we're denoting a set a

list of things in mathematics we use the

curly braces to show here what we have

as elements and our set and our elements

are separated by commas inside of the

curly braces another way that we might

represent these sets instead of writing

them out one element at a time as we

might use what's called set builder

notation so remember we said set a was

going to be the weekdays and set B is

going to be our weekend days here our

set a in set builder notation we have

our curly brace to start the set and

then we say X so this is the set of all

things we're calling X this line tells

us that the rule for X is coming after

so the set of all things we'll call X

such that X is a weekday our set B is

just defined as all objects such that

our object is a weekend today so instead

of listing the elements we might use set

builder notation that's very handy

especially if the set has a lot of

elements in it and we don't want to have

to list them all out individually if we

want to talk about something being an

element in a particular set then we'll

use this little rounded looking object

so this says that Thursday is an element

of a it is in set a you can see that

here obviously Thursday is a weekday

Thursday is if we have the is an element

of with a slash through it that's the

same as like not equal so here we are

not an element Thursday is not

element of set beat when we have the

slash through it let's say we define two

other sets let's say I define the days

that I have class are set C and I have

class on Monday Tuesday Thursday Friday

each week and let's say that you you

probably don't have class these days but

I'm just making something up days that

you have class let's say or set D and

those are Monday Wednesday Friday

Saturday so we might talk about the days

that we both have class and that's

called an intersection so here we read

this notation as C intersect D or the

intersection of sets C and D and the

intersection is simply all of the

elements that are in both sets C and set

D to be in the intersection of something

you need to be in both sets so if I want

to figure out what the intersection of

sets C and D are I look for any day that

is in both lists I noticed that Monday

is in both lists and I also should

notice that Friday is in both lists and

those are the elements in both lists so

we would say the intersection of sets C

and D is Monday and Friday those are the

elements

the Union so we represent this the other

way this looks like a you right instead

of an upside down you so this is the

union of sets C and D and a union is

when you have an element in at least one

of the sets so it's in either set C or

it's in D or it could be in both right

as long as it's in at least one of the

sets that we're listing here so if we

think about all of the elements here all

of the days that are in either C or D or

both we just look for anything that is

named in the collection here so Monday

is named Tuesday is named here Wednesday

is named here Thursday is in C so it

counts Friday as in D and C being in

both as okay to Saturday the only thing

that we don't have is Sunday so it looks

like Monday through Saturday are the

elements in our C Union D remember it's

just required to be in at least one of

them it can be in both let's go back to

our sets a and B where a is the set of

all weekdays and B is the set of all

weekend days so we want to think about

our intersections and unions using these

two sets so here this says what is the

intersection of a and B in other words

what elements are in both a and in set B

well these are all weekdays and these

are weekend days so there actually

aren't any elements that are in both of

the sets all of Monday through Fridays

and a Saturday Sundays and B so there

are no elements actually in both of the

sets so we would have no elements in

this a intersect to be the intersection

of a and B when we have no elements in a

set we call that the empty set we write

it with a circle with a little diagonal

slash through it that's the notation for

the empty set and that's the set that

just doesn't have any elements in it so

if you say you know what elements are in

both well there are none so that answer

is the empty set another way that will

say that sets have no overlapping

elements is that we'll say the sets are

disjoint if they have no elements in

common let's look at this bottom one now

we have a union B so the union remember

that just means it needs to be in one or

the other

or it can be in both so if it can be in

one or the other that means all the

weekdays are going to count that means

all the weekend days are going to count

as well and that means everything that

we could possibly consider right all

seven days of the week are going to fit

in the union here when we talk about a

set that has every possible element that

we could be considering in that moment

we call that the universal set so

because a union B has all seven days of

the week that's considered our Universal

set for this type of a situation going

back to our sets C and D we want to

illustrate the complement so the

complement is usually denoted with a

little apostrophe next to the set aim so

this is actually read here the

complement of set C and the complement

of a set is simply all elements in the

universal set so think about all the

things we could possibly be talking

about what of those objects are not in

set C and since our universal set is

days of the week when I look at set C I

think about well what are the days of

the week that are not in set C I notice

I don't have class on Wednesday I also

don't have class on Saturday or Sunday

so my C complement is going to be

Wednesday Saturday Sunday if I look at D

complement remember D is the days that

you have class so D complement would be

thought of as the days that you do not

have class in other words what days are

not in set D and in that case you don't

have class on Tuesday Thursday and also

on Sunday according to my list here the

last basic thing we want to introduce

you to with sets and set operations is

called a sub set so we have some set e

is a subset of another set F if any

element in E is also an F so everything

in this one is also in this one it kind

of fits inside of it so to speak then

that means that E is a subset of F this

looks similar to like a less than equal

to

but you'll notice it's actually around

you so this says E is a subset of F in

other words everything in E is also in F

so my Universal set all of the objects

that we're thinking of are just all the

days of the week remember a was my set

of all week days B was my set of the

weekend days C that was the set of days

I had class and D that was the set of

days that I said

had class whether I was telling the

truth or not I suppose so just

illustrating some of this subset stuff

here this says that C is a subset of a

is C a subset of a it is right because

Monday Tuesday Thursday and Friday all

of those elements are also in set a

Monday is an A Tuesdays and a Thursday

is an A and Friday is an A so everything

in C is also an A and we say C is a

subset of a we can also talk about

something not being a subset of another

set like we said something was not an

element of a set so here if we put a

slash through the subset symbol this

says that D is not a subset of a ok if

we think about D being Monday Wednesday

Friday Saturday Monday is an a Wednesday

is an a Friday is an A but Saturday is

not in a so many of the elements of D

are and a but not all of them and since

not all of them are also an A then we

say that D is not a subset of a one

little additional thing that we'll

mention is that the empty set is a

subset of any set technically we think

of it that way because when we say

something is a subset we say any of its

elements are also in this other set and

since the empty set has no elements that

can be missing from another set then

technically all of its elements are in

any other set that we look at right D

failed to be a subset of a because we

had Saturday in here and it wasn't in a

and so if we were going to say the empty

set is not a subset of some other thing

there would have to be something in the

empty set that was missing in the other

one and since there's nothing in the

empty set then we can't have that happen

so the empty set is considered to be a

subset of anything of ABCD you any set

that we can think of ok everyone

hopefully this helps you with sets and

set operations getting started with some

of these symbols and how to interpret

them thanks for watching we'll see you

in the next video