Welcome to Block 2 video.
This is about linear algebra.
In this first video, we will see basic definitions
about sets and operations.
Cartesian product and we will see some applications.
I am Lidia Montero
and I will be your instructor for this video
and the companion video with exercises
Introduction.
What is a set?
A set is just a list of elements.
So the names of the students in a class,
the name of the courses
that are offered in the Master of Urban Mobility.
The least of the states in the United States, etc..
Okay. Order does not matter.
So it means that you can use any order
What is important is the elements
Set notation is
useful and is very useful to understand how to operate
with data frames and statistic software.
So let's continue. First
remark is that sets 24
do not have duplicate elements so
if we have digits from 0 to nine.
Then if you repeat several times three for example,
as here as you can see in the screen then the set
is just the same as the one that has been written below
because we have to remove all duplicated.
So it wouldn't make sense to repeat
the name of the courses in the Master of Urban Mobility.
So, no sense
The difference between sets and lists.
depends on the
the framework and the sector we are working with.
But in general, in linear algebra we are using,
we are dealing with sets and we do not use lists. In programing,
we are using lists, not sets, in general.
Okay, so let's move forward to notation.
The notation for sets are capital letters.
Okay.
The easiest way to specify a set
is just by enumeration of all the elements.
But then take it into account that the number of elements
can be finite or infinite.
Then it's sometimes useful to have a notation that it is a
builder notation.
For example, as the one that I am showing in the screen
A contains all values
that are even and greater than four.
So, the vertical bar has the meaning
of such that satisfy a condition.
For example, in B I have such
that x is odd and greater than seven.
So, this is how we notate the sets
and let's see the common sets.
So, we have seen that the set contains
elements, elements in a given domain,
sometimes an individual element, an individual
item, as for example, S, it is
important to identify whether
it belongs or not to a given set, capital S.
So, let me in the set of all digits
between 0 and 9, 5
would be an element of this set.
In the case of a set
that is three, five, seven, nine
then 1 wouldn't be an element of this set.
Okay use and pay attention to the notation
because we use this strange E letter so belongs to
is the E and with a slash
then it means 'it does not belong'.
We have examples with the
set builder or just the enumeration.
But in this case, I choose the enumeration point of view.
Then the set of natural numbers, i.e.
1, 2, 3 to infinite numbers. The set of the integer numbers
from minus infinite to -2, -1, 0, 1, etc.
and all the positive values.
Then the set of integers that are positive.
So one, two, three, four, etc.
The set of rational numbers.
So numbers that can be written as the quotient between two integer numbers.
Okay.
So in this case we have seen some examples
about numbers that are common in linear algebra. Now,
let's continue with the
the concepts and also the operations.
Two sets are equal if they have the same elements.
So, since order does matter, in the first example
the set 1 to 5 is the same
as the set 5 to 1.
If we have duplicate elements
these duplicate elements do not account.
So, in the second example
we have the same set.
Two sets are not equal
if they do not have the same elements.
Okay as in the example and it's important the concept of a set
S that is included into a set T
So, 'include' uses
a special notation as you can see on the screen
and in the example of S={1,3,5}
and T that is 1 to 7, but
since all the elements in S are elements in T
Then I say that S is included into T.
If a set
S does not contain one or several elements
that belong to T, then 'sorry'.
S set does contain elements that do not belong to T.
Then in this case, S is not included into T.
The empty set is a subset of all sets.
And all sets are subsets of the universal set.
So the universal set will be seen in the next slide.
First of all, let's focus on the notation
about the cardinality, the number of elements in a set.
This is usually written with
the name of the set between vertical bars.
Okay, we have the concept of the power set.
The power set of S is the set of all possible
subsets included into S.
We have the example of S
that would be an experiment of tossing a coin.
And then, we have two possible outcomes tail and head.
Okay, so the set,
the power set with all possible subsets would be written
as capital P of S and the possibilities are
the null set, then
{T} second subset, then
{H} that is the third subset.
And finally {T,H} that
is S set, S itself.
So, we have that the cardinality of S is 2
and the cardinality of the power set of S is 4.
And
this rule can be generalized.
So you we have a set with n elements, then the power set
will contain two squared elements.
So, that is 2 raised to the n-th power elements.
What is the universal set?
It is in a given domain, the set that containes all the possible elements
Okay? So for example, in the case
of the set of students in a class, in a course,
then the universal set would be the set of
all of students in the university.
Okay.
So, we have several examples.
Pay attention at the notation and do not confuse the notation of this U
that is in cursive and bold
and this is just to establish
a differentiation between the operator, the union operator,
and the name of the universal set.
Okay.
So sets can contain other sets
no matter and the null set
is the set that contains no elements.
Okay, so an empty set is always
an element of a given set and it is not the same
the null set that the set that contains the null set.
Okay, this is a little bit technical.
If we move forward into
the graphical representation of sets
we have Venn diagrams.
Venn diagrams are very useful to understand some of the operations in
sets, so let the rectangle be U, the universal set
and then I have a circle, a set, that contains the vowels in the alphabet.
So, the universal set is the set of the letters in the Latin alphabet.
Then S contains the vowels
and the rest are not vowels.
Usually the single elements are not indicated or rounded with circles.
So, in
this case we have a graphical representation
and this will be very useful to understand
the operations. But, before entering into the operations.
I would like to define the Cartesian product.
The Cartesian product is composed
of the pair of ordered elements
so that the first element is from the first set and the second element
is from the second set, and these are ordered pairs
belonging each of them, the first and the last to one of the sets.
So the notation is A x B.
For example,
if I have A that is {a, b}, so A contains two elements
and I have to set B that contains 0 and
1, the Cartesian product would be just all possible pairs,
ordered pairs and in this case I use parentheses
to indicate that the first pair could be (a,0).
Then I have (a,1), then (b,0).
And finally (b,1).
So these C is the set
containing the Cartesian product between the two sets.
Okay, so formally it's
just the pairs where the first element belongs
to one of the sets, and the second element belongs to the second set.
Let's move to set operations.
And firstly, the union.
We have the universal set and two sets
included into this universal set that are A and B,
So the A union B is
the set that is composed of the x elements
where x belongs to A
or x belongs to B.
Graphically, it is the grey area
that you see on the screen.
We have some properties, basically the identity,
a set union with
the null set is the same set.
We have also the domination law, the union of A and the universal set is
the universal set.
The idempotent law,
A union A is the same A
and the properties of the commutative law.
So referring to the order that we apply in which
we write the two set and also the associative law.
The next operation is the intersection.
Intersection is composed
of the elements belonging to the universal set
that belong to A
and simultaneously belong to A, so graphically,
it is the area that is shared by these two sets,
that are A and B. Properties.
We have also some properties and the properties
I would like to focus on are: the commutative law.
So, order does not matter in this operation. On the associative law.
Okay, We say that
the two sets are disjoint
if they do not share any element,
so, the intersection is the null set.
And then let's see the complement.
The complement for a set
B consists of all the elements belonging to the universal set
that do not belong to B.
The notation is B with the bar
on the top and
the complement of B is the grey area that you see.
So, in fact, it is the universal set
minus B. Interesting properties: basically, twice
the complement is the original set.
The union between one set
and the complement is just the universal set.
The intersection between one set
and its complement is the null set.
And finally, I have to refer to
what are known as DeMorgan's Laws.
In the DeMorgan's laws, we establish a relationship
between the intersection and union operators.
So, let me start by the first and the complement of the intersection
is in fact the union of the complements.
And the second one that would be the complement of the union
is just the intersection of the complements of the two sets involved.
These properties can be generalized to go with more sets, I have to say.
But the critical point is the combination
between the basic set operator and the complement.
And these are nice references
and this has been the last part
of this introduction to sets.
Thank you very much for your attention.