INTRODUCTION to SET THEORY - DISCRETE MATHEMATICS
Summary
TLDRThis 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
- ๐ข Sets are collections of objects, known as elements, which can be represented visually or in curly braces.
- ๐ Sets can be finite, like the numbers 1, 2, and 3, or infinite, like the set of all positive integers.
- ๐ Elements in a set are unique, meaning repeated elements are only listed once, and the order of elements does not matter.
- ๐ 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).
- ๐ 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.
- 0๏ธโฃ The empty set, denoted as {}, has a cardinality of zero as it contains no elements.
- ๐ The set containing only the empty set, {{}}, has a cardinality of 1 because it has one element: the empty set itself.
- โ๏ธ Set builder notation allows for a more formal representation of sets using rules and conditions, such as {x โ Z | x < 6} for all positive integers less than 6.
- ๐ฆ Sets can contain other sets as elements. For example, a set containing an empty set and another set has two elements.
- ๐ง 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 (โ) to denote that an element is part of a set, such as 'a โ 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 ร 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 โ 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
๐ 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.
๐ข 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.
๐ 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.
๐งฎ 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.
Mindmap
Keywords
๐กSet
๐กElements
๐กFinite and Infinite Sets
๐กOrder
๐กCardinality
๐กNatural Numbers
๐กIntegers
๐กRational Numbers
๐กEmpty Set
๐กSet Builder Notation
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
5.0 / 5 (0 votes)