Original subtitles

Instructor: Hi there.

In this lecture, we are going to explain

what a union of two events is and how we can use it.

Okay, now we know which part of the graph represents

the intersection of A and B

but what if we only require one of them to occur?

Regardless which one.

That is the same as asking either A or B to happen.

In such cases, we need to find the union of A and B.

The union of two sets is a combination

of all outcomes preferred for either A or B.

If you think about different American states

as separate sets then the citizens of each state

would be their elements.

Just like every Texan is an American,

so is every element of an individual set

also an element of the union.

We denote the union of two sets as A union B,

the symbol we use is just the intersection symbol

flipped upside down.

To avoid confusion between the two,

remember the symbol for union resembles a capital letter U.

Okay, let us examine what the unions would be

in the three different cases.

If the sets A and B do not touch at all

then their intersection would be the empty set.

Therefore, their union would simply be their sum.

Intuitively, this makes sense.

No element is in both sets simultaneously

so we do not need to worry about double-counting.

Going back to the card example,

the union of hearts and diamonds would be all red cards.

No card can have multiple suits

so we need not worry about counting a card twice.

Therefore, the number of red cards equal the union of cards

which are either diamonds or hearts.

If the events intersect, the area of the union

is represented by the sum of the two sets

minus their intersection.

That is because if we simply add up the area

of the two sets, we would be double-counting

every element that is part of the intersection.

For instance, five people in the office have blonde hair,

four people have blue eyes, and only Kate has both.

Therefore, there are only three non-blonde people

with blue eyes and four blonde people with non-blue eyes.

Thus, the union of blonde and blue-eyed colleagues

is the sum of people who have

precisely one of the two features

as well as Kate who has both.

Mathematically this result is three plus four plus one

or eight employees, satisfying at least one

of the conditions.

Alternatively, we can compute the union using the formula

we mentioned earlier.

We have five blondes plus four blue-eyed people, minus one

or a total of eight colleagues who possess

at least one of the two distinct features.

In this instance, the minus one represents Kate

who is the intersection

since she is both blonde and blue-eyed.

We can use the notation we learned so far in this course

to express this.

The union of A and B equals the sum of the sets

of A and B minus their intersection.

In fact, the union formula we just showed you

is universally true

regardless of the relationship between A and B.

Okay, so what happens if B is a subset of A?

Well, in that case, the union would simply be

the entire set A.

Imagine event A is being from the U.S.

and event B is being from California.

If you talk about all the people

who are either from California or the United States,

you are simply talking about all the people from the USA.

Remember that the intersection of these two sets

is equal to the entire set B.

So the intersection of A and B

represents all the people from California.

If we plug this into the formula from a few minutes ago

that would give us the following statement.

The union of all people from California or the United States

is the sum of all California natives

and all Americans minus anybody born in the Golden State.

After canceling out all Californians

we get exactly what we expect.

The union of people from either California

or the United States equals the entire population

of the USA.

Now that we know what unions and intersections are,

we can talk about mutually exclusive sets.

In the next lecture

we are going to define the term mutually exclusive

and discuss certain characteristics of such sets.

Thanks for watching.