In this video, we are going to introduce you
to the notion of intersections.
Let’s start with the intersection of two events.
When we want both A and B happening at the same time, we are talking about their intersection.
Graphically, the “intersection” is exactly what the name suggests – the area where
these events intersect.
It consists of all the outcomes that are favourable for both event A and event B simultaneously
and we denote it as, “A, intersect, B”.
Let us look at the and see how the intersection changes.
The intersection of all hearts and all diamonds is the empty set.
The empty set has no elements, so there are no outcomes which satisfy both events simultaneously.
We would write this as: A intersect B, equals the empty set.
Now you see why we needed this notation!
Let’s consider a different example.
The intersection of all diamonds and all queens is represented by the Queen of Diamonds.
That card is the only one that satisfies being a queen and being a diamond at the same time,
so nothing else is included in the intersection.
In the example with red cards and diamonds, the intersection of the two would simply be
“all diamonds”.
That is because any diamond is simultaneously red and a diamond.
We would write this as: “A intersect B, equals B”.
Great job, everybody!
We use intersections only when we want to denote instances where both events A and B
happen simultaneously.
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 that 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 equals 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 a part of the intersection.
For instance, 5 people in the office have blond hair, 4 people have blue eyes and only
Kate has both.
Therefore, there are only 3 non-blond people with blue eyes and 4 blond people with non-blue
eyes.
Thus, the union of blond 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 results in 3, plus 4, plus 1, or 8 employees satisfying at least
one of the conditions.
Alternatively, we can compute the union using the formula we mentioned earlier.
We have 5 blonds plus 4 blue-eyed people, minus 1, or a total of 8 colleagues who possess
at least one of the two distinct features.
In this instance, the “minus 1” represents Kate, who is the intersection since she is
both blond and blue-eyed.
We can use the notation we learned so far in the 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 US” 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 cancelling 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.