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(male narrator) So consider these three sets:
A, the set of all even numbers;
B, the set 2, 4, 6; and C, the set 2, 3, 4, 6.
Are each of these statements true or false?
Now to understand this,
we're gonna need to talk a little bit about sets.
So a set is simply a collection of objects,
uh...that contain elements of that set.
There's two ways to describe sets.
One is this way, using a sort of verbal description
or a description of what the set contains.
And then the other is using this notation,
where the curly brackets,
uh...indicate we're going to list some stuff,
and then this lists the contents of the set.
Now it's also possible to have a set
that doesn't contain anything,
and that is called the "empty set,"
which is usually denoted like that.
Uh...but that doesn't come into this particular problem.
So-so the first question here asks,
"Is this true?"
Now to understand this,
we're gonna need to know what this symbol is,
and this symbol means "element of."
So this says, "2 is an element of B."
In other words, is 2 an element of the Set B?
Well, the Set B contains 2, so yes, this is...this is true.
Now if I was to ask,
is, you know, 7 an element of the Set B?
Then of course, that would be... that would be false.
Now the second statement here,
uh...again, we need to know what the heck this symbol means.
Uh...so this symbol means...means "subset of."
Uh...and more correctly, it is a "proper" subset of.
Now the idea of a subset is that
if you have a-a bunch of items,
uh...let's say here is-is a big set of items,
then a subset...
is a smaller collection
that contains elements from the larger set.
That's the idea of a subset.
Uh...so if we were to look at the Set A here,
A is the set that contains all even numbers.
So that's 2, 4, 6, 8, 10, 12, so on and so forth.
So is B a subset or is it contained within the set of A?
And you'll notice that the elements from B--
2, 4, and 6--all do, in fact, exist inside Set A,
so this is-is true.
Now if we were to ask, let's say, is B a subset of C?
Even though some elements of...oh, actually,
it looks like all of the elements of B do exist in C,
and so this would also be true.
Now if we asked, "Is C a subset of A?"
This is false,
because C contains an element that is not part of A,
and therefore, uh...it is not a subset.
Now I called this a proper subset,
because there's a second notation,
which looks like this...
and this is just the generic subset of.
Uh...this little line underneath it implies equality.
Kind of like a less than and equal to sign.
Uh...so if we could say that B is a subset of B,
because they actually end up... or they're actually equal.
Uh...and so we could correctly write this.
It would not be correct to write B is a proper subset of B,
because they're actually equal.
So that's not true, but this is just fine.