Basics and Vocabulary

Basics and Vocabulary

The goal of this hour is to give you the basic vocabulary you need to understand
more advanced material later in your mathematics studies. We begin with the
definition of a set, then discuss a fancy word called cardinality, which is really just a
fancy way of saying "size." We then go over two ways to combine sets–intersection
and union. All this will be presented in a dry manner at first, but there is light at the
end of the tunnel.

So, what is a set?
The term set refers to a collection of objects, called its elements. Sets are denoted
using braces or brackets and it is possible to have elements that are themselves
sets. For example, the set A can be written as {a}, whereas the set E could be
written as {apple, monkey, DE}.
Sets are collections of elements. In this example, A is a set with four elements: the
element one, the element two, the element minus three and the element seven.

E is
a set with three elements: apple, monkey and DE. The elements in a set can be
essentially anything. We use braces to enclose sets, so that we can see which things
are contained within them. For example, when we write two inside A – without the
braces – that means two is an element of A. Minus three is also an element of A:
likewise for eight, which is not an element of A. So we write "not in" using this symbol
here – it means exactly what it says on the tin: eight is not in A.

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The last notion we'll talk about is that of cardinality. Cardinality is the size or number
of elements in a set. When we refer to the absolute value of a set, we use this
notation: A]. So in this particular case, where A is equal to one, two, minus three,
seven, the cardinality of A is four, and the cardinality of E is three. IS That's all there IS
to it!
Let us move on to the next topic. We are writing three sets for you here. A is equal to
one, two, minus three, seven
as you
have seen before. Let us say B is equal to
two. eight. minus three, ten, and D is equal to five and ten. Now you will notice that
of these three sets, they share some elements in common (the intersection), some
elements they do not share in common (the union). There are two concepts called
intersection and union which really allow us to talk about those more rigorously.
First, let's consider the intersection of sets A and B. The intersection of these sets is
the set of elements that are contained in both A and B. In other is words, it is the set of
elements shared by both sets. To determine the intersection of sets A and B. we
must first determine what elements these sets have in common. We see that two is
contained in both A and B, as well as minus three. Since these are the only elements
contained in both sets, their intersection equals {2, -3}. Next let us work out the intersection between set B and set D. This intersection is defined to be the set of
elements that both B and D contain in common; however there are none so this set
is empty or {. The notation for an empty set is a boldface zero with a slash through
it: 0/. This is called the empty set. Hard to spell. And by convention, we always say
that the cardinality of the empty set is zero – there's nothing in it.
[Graph]
Here is a simple example of a set that can be defined in multiple ways. Another way
of writing A intersect B is to give you a recipe for computing it yourself. So the
definition of A intersects B, instead of listing out the elements, we can list it this
way:
it's a set of x, we don't know what x is, but now I'm gonna give you conditions that x
satisfies, that's what this little colon here means. The set of x such that x is in A and
x is in B. This notation here is very, very important and we're gonna It see it over and
over again.
You can think of "A intersect B " like membership in a club. The club has two rules:
(1) you must be in A and (2) you must be in B. If any x satisfies both conditions, then
the bouncer will let x into the club. If no x satisfies both conditions, then no x gets
into the club. For example, suppose two people come along and say: "I want to get
into this music club." The bouncer checks his ID and says "okay"; so two get in.
Suppose one comes along: "I want to get into this music club." The bouncer checks
his ID and says "no"; so one does not get into the club.
Let's rewrite those sets and give you a different definition. So A is equal to one, two,
minus three, seven; B is equal to 2, 8, -3, 10: and D is equal to 5, 10. The next idea
we're gonna define here is the idea of the union. A union B is read as the set of
elements that are in A or in B or in both. The symbol for union is a rectangle with a
smaller rectangle inside it. When you see A union B, think about it as being inclusive
of elements from A and from B. In this example. vou have 1. 2 and 5 from set A and
7 from set B. So the answer is 8, 10. In another notation, A union D would mean 1, 2
3. 7 and 10.