All language subtitles for 001 Sets and Events_en

Narrator: Welcome back.

In this section of the course,

we are going to learn how to describe events

and the ways in which they interact with one another.

To do so, we need to introduce a few concepts.

Let's begin.

Every event has a set of outcomes that satisfy it.

These are the favorable outcomes we mentioned earlier.

For example, the event could be being even,

and the set of values would consist of two, four, six,

and all other even numbers.

However, values of a set don't always have to be numerical.

For instance, an event can be being a member

of the European Union.

Values like France or Germany would be a part of this set,

and values like USA or Japan would not.

Convention dictates that we use uppercase letters

to denote these sets and lowercase letters

to express individual elements.

In the numerical example from earlier,

uppercase X will express all even numbers,

and lowercase X would be a single value of that set,

like eight.

Great, you should know that any set can be either empty

or have values in it.

If it does not contain any values,

we call it the empty set or null set,

and denote it with what looks like a crossed-out zero.

Let's focus on non-empty sets for now.

Non-empty sets can be finite or infinite,

depending on the number of elements they have.

When working with them, we often want to express

if an element is part of a set.

The symbol we use to denote that is the E symbol.

We read it as "is an element of," or simply, "in."

For instance, the following statement reads as "x in A,"

meaning x is an element of the set A.

If we want to state that the set A has an element x,

we just change the order and flip the symbol 180 degrees.

The new expression reads as "A contains x."

Having two ways to express the same thing

may seem redundant, but it may prove useful

for you in the future.

Right, but what if we wanna show

that an element is not contained in a set?

Well, we can use the same notations,

but simply cross out the E symbol

with a single diagonal line, like so.

The statements now mean x is not in A,

and A does not contain x.

Okay, great.

In other situations,

we would need to make generalized statements

about multiple elements within a set.

To do so, we introduce the symbol representing a capital A

turned upside down, which we read as "for all" or "for any."

For instance, if we want to make a statement

regarding the entirety of a given set,

we simply write for all x in A.

Another important symbol is the colon.

We often encounter a colon after using the "for all" sign.

This colon reads as "such that."

It is incredibly useful when we wanna make statements

about a specific group of elements within a set.

For example, if we wanna state something

that only concerns the even numbers in a set,

we can write for all x in A, such that, x is even.

Great job so far, everybody.

The final fundamental term we must introduce is subset.

In short, a subset is a set

that is fully contained in another set.

If every element of A is also an element of B,

then A is a subset of B.

We denote that with A subset B.

As you can see, not all elements

of B are necessarily part of A.

Going forward, remember that every set contains

at least two subsets, itself and the null set.

Good job.

Now that we have introduced some of the basic notation,

we are ready to explore how different events interact.

In the next video, we are gonna show you how

to visualize the relationships between two or more events.

See you there, and thanks for watching.