> Home > Statistics Notes > Probability > Sample Space

## Set Theory

An experiment is a collection or set of outcomes. The set of all outcomes of an experiment is called the sample space; it is common to use a capital Greek letter omega to represent the sample space. The capital omega looks like this: . The symbol is a lower case omega; it will often be used to represent some particular element of the sample space.
If is a member of the set , then we write . This is pronounced "little omega is an element of [the set] big omega", or "little omega is in [the set] big omega". A given element is either in a particular set, or it isn't; the concept of being in a set multiple times isn't defined.

## The Empty Set

Suppose that a certain set has no elements. Such a set is called the empty set, and can be written {}, or . These represent the same set and have the same meaning.

## Subsets

Suppose A is some set, and that whenever is a member of A, then it is also a member of another set B. In other words, every member of A is also a member of B, or put yet differently, B contains all the members of A. Then A is a subset of B.
Example: In the die example, the sample space ={1,2,3,4,5,6}. If we define a new set, A={2,4,6}, then A is a subset of , since every element of A is an element of . We write this as A . We can also define another set B={4,6}, then B is also a subset of . Moreover, B is also a subset of A as well. But C={1,7} is not a subset of , because the element 7 is present in C, but it is not present in .
If we have any set A, then we may say that it is a subset of itself: A A. A is a subset of A, because every element of A is an element of A. Also, we say that the empty set is a subset of every set; the only way the empty set could fail to be a subset of any set would be to have an element that the other set didn't have. But the empty set has no elements, so it cannot fail to be a subset of any set at all.

## Equality of Sets

Suppose that A B, so that every element of A is in B. And also suppose that B A, so that every element of B is in A. Then A and B have to be the same set, since they consist of the same elements: A=B. This is how the mathematicians defined the equality of sets.

## Proper Subsets

Suppose once again that A B, so that every element of A is in B. But now suppose that it is not true that B is a subset of A; there are elements in B that A does not have. For instance, if A is the even integers and B are all the integers, then we know that A is a subset of B, since every even integer is an integer. And also there are integers that are not even, so B is not a subset of A. Whenever A B, but A does not equal B, then A is called a proper subset of B. In the example, the even numbers form a proper subset of the integers.

On to events.