Discrete Math - 2.1.1 Introduction to Sets
Summary
TLDRThis 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
📚 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.
📐 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.
🔍 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.
Mindmap
Keywords
💡Set
💡Element
💡Natural Numbers
💡Whole Numbers
💡Integers
💡Rational Numbers
💡Real Numbers
💡Complex Numbers
💡Roster Notation
💡Set Builder Notation
💡Interval Notation
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
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