# 1.1 Sample sets

## Abstract

Today we begin our new category of blogs, and they are about probability theory. To begin with, I have to say that the knownledge in these blogs are elementary, while, that just means they are simple and basic but not unimportant. However, these concepts in the following blogs are foundations of the whole probability theory.

The first bolg contains basic concepts of “Set Theory”, and the probability theory is built on them.

**keywords**: *Set, Point, Space, Sample Point,Sample Space, Subset, Empty Set*

## The Map of Concepts

## Sample Sets

A set of things or a bunch of things means that several things are collected together. For instance, we know a bunch of super heroes of Marvel, and they have a team called “Avengers”, here is some of them in early story:

Frome left to right, they are:

1. “Scarlet Witch” Wanda Maximoff

2. “Quicksilver” Pietro Maximoff

3. “Thor” Thor Odinson

4. “Iron Man” Tony Stark

5. “Hulk” Bruce Banner

6. “Captain America” Steven Rogers

7. “Black Widow” Natasha Romanoff

8. “Hawkeye” Clint Barton

Well, If they wanted me to join them, the photograph would be changed to:

Aha, this new team looks more powerful. And there are 9 heroes, the the 9th is “Super Handsome Boy” Tony Tan.

But now we have a problem: Can you list the *set* of gentlemen in the picture? That’s so easy, right? The answer is:

“Pietro, Thor, Tony, Bruce, Steven, Clint, and Tony”

Wait a moment, this answer looks correct, but it’s not so precise. That’s because there are two Tonys in the list. And according the list, we can not discriminate them. In other words, we have no idea about the first Tony is the rich guy or the handsome guy and vice versa( I am the handsome guy! 😆 )

Well, this question become a set theory question, now. The heroes in the picture are the “a bunch of things”, we call it a set. **In a set, A same member can never appear in a set more than once.** So the “Tony”s in the list is not just making us confusing about who is who, but also make the list not a set.

The right way to represent the guy’s name in the “set” way is:

“Pietro M, Thor O, Tony S, Bruce B, Steven R, Clint B, and Tony T”

In the new way, we can easily discriminate between “Tony S” and “Tony T”, which the first guy is the rich one, and the second one is the handsome one. While this is a good way but not the only way to make that clear, we can also use their full names or their nick names to represent them.

There are more examples in mathematics about sets:

1. Prime Numbers: the set of integers, which have no other divisors except 1 and itself.

2. The interval $(a,b)$: the set of real numbers, who is bigger than $a$ and less than $b$

3. Circle: a set of points, who are equidistant from a given point.

### A Bunch of Things

All the examples above are about “set” in mathematics. However, in probability theory, the notion of a set plays a more foundamental role. In probability, we are interesting in general kinds of sets as well as specific concrete ones. Here is some examples:

(a) a bushel of apples

(b) 55 cancer patients under a certain medical treatment

(c) all student in a college

(d) all points on a target board

And with these four examples, we can get four *smaller* sets:

(a’) the rotten apples in the bushel

(b’) those patients who respond positively to the treatment

(c’) the computer science majors of the college

(d’) the points in the “bull-eye” area on the board

All the things we list upside are in our daily life. If we want research those in a mathematical way, we have to convert them into mathematical form. Mathematical modeling is the process of using mathematical structures to represent the real world situations, and all the examples above can be described by set theory.

### Set, Space and Points

We call the things in the examples: *Point* , so an apple in the buchel if apples is a point. Any one cancer patient in the 55 patients is a point. A student in the college is a point, too.

Point is one thing in the bunch of things, and the bunch of things also has a special name — “Space”. According to these definition, all the apples of the bushel of apples are the space; All the 55 cancer patients is the space; All the students in the college is the space; And all the point on the board is the space.

We have learned too much concepts about “point” or “space”, like in geometric , physics, and other mathematical subject. To distinguish from the other “space” or “point”, we prefix them by the word “sample”. So, in set theory, we developed the concepts of “Sample Point” and “Sample Space”.

The sample space consists of a number of sample points, and is just a name for the totalty or aggregate of them all.

All the examples (a-d) and (a’-d’) are all sample space.

Smaple Space is denoted by

$$

\Omega

$$Sample Point is denoted by

$$

\omega

$$

They both are Greek letter, and read as *omega*. Because we always have many points, we can add subscripts or dashes, such as: $\omega_1,\omega_2,\dots,\omega’$

Since we have fixed the $\Omega$, we will just call it a set.

## Subset

As example (a’-d’) is a part of example (a-d), and we have just fixed the (a-d) as set, So the (a’-d’) is a subset.

### Empty Set

In extreme case, the subset can be the set itself or can be an empty set, where there is nothing in it. Although the empty set looks so weird, it is a very important entity and it has a special symble:

$$

\emptyset

$$

### Size of a Set

In the super hero example, those heroes are going to eat something, for example, they go to the Mcdonald’s and have some hamburgers. If each of them eats only one a time, how many hamburgers do they need. The answer is the number of people – 9, of course, include me.

The people are the set, and the number of people is also the size of the set. So, the size of the set is the number of the points, and when the Super heroes set is denoted by $S$, the size of the set is denoted by:

$$

|S|

$$

The size of set must be an integer and more than 0. And the set can contain $\infty$ points, So, the size of a set is an nonnegative integer or $\infty$

## Relationship between Points and Sets

### Belong or Not

The sample set or sample space is filled with sample points. When a particular set $S$ is well defined, we can tell wheter any given point belongs to it or not.

When the point belongs to the set, it’s denoted as:

$$

\omega\in S

$$

While, if the point does not belong to the set, it’s denoted as:

$$

\omega\notin S

$$

One can always quibble about the meaning of words such as “a rotten apple”, “respond positively” and so on. However, we will not indulge in them here.

A set must have certain points, in other words, a point can never be in and not in a set at the same time. So the definition of a set become important. One way to specifying a rule to determine a set is to **enumerate all its members**, to make a complete list like the super heroes example. But this is too tedious we the members are more than ten, such as list all the students in the college may be possible, but it is not economy.

A new example: we throw 6 dice in a game, there are $6^6$ kinds of results ( the results can be denoted as set $S$, and the size of results $|S|=6^6=46656$ ). We can never list all of them out. But we can describ them in a systematic and unmistakable way as the set of all ordered 6-tuples of the form:

$$

(s_1,s_2,s_3,s_4,s_5,s_6)

$$

We have 6 dice here, so each of them produce the results $s_j$ where $1\leq j\leq 6$ and $s_j$ can be $1,2,3,4,5,6$. This is a good illustration of mathematics being economy of thought (and print space).

## Relationship between Set and Set

Example (a’-d’) is smaller than (a-d), do they have some relations?

### Inlucude and Contain

If all the sample points of the sample set $A$ are in the sample set $B$, we can say $A$ belongs to $B$, or $B$ contains $A$. And, of course, $A$ is a subset of $B$, this relation can be written as:

$$

\begin{aligned}

A\subset B && B\supset A

\end{aligned}

$$

### Identical

If two sets have all the same points, they are indentical:

$$

A=B

$$

Two ways to testify two sets are identical:

1. Test every point in set $A$ to make sure they are all in set $B$, the test every point in set $B$ to make sure they are all in set $A$

2. Try to proof $A\subset B$ and $A\supset B$

The seconde way is a roundabout method, but in some conditions it is the only way to check if they are identical. Sometimes the identical sets look more different, such as the set of even and the set of solution $x$ of the equation $\sin(\frac{\pi

x}{2})=0$. They are identical, after several calculated steps.

## Conclusion

We learned some concepts of Set theory, they are the foundations of probability theory. Sample space, sample points are the most important concepts, so are their relationships.

## Reference

- Chung, Kai Lai, and Farid AitSahlia. Elementary probability theory: with stochastic processes and an introduction to mathematical finance. Springer Science & Business Media, 2012.

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