# Sets

As we start to get into probability and statistics, we will hear the term "set" used frequently. But what exactly *is* a set? The definition is probably much simpler than you realize, and it's definitely not hard to wrap our minds around this concept. That being said, operations with sets can be very complex in some cases.

## What is a set?

A set is a collection of events, numbers, names, qualities, or anything else for that matter. A set often represents the result of an experiment. For example, we might measure how many times our teacher wears the color blue each month. Or perhaps we might measure the number of ice cream cones we eat each day.

A set can also be a collection of labels rather than numbers or results. For example, our "set" could simply be a list of our friends:

{John, Clarence, Mika, Ferdinand, Veronica}

Note that we "contain" our set within these symbols "{}." These symbols are called braces.

You might also see a set of numbers: {2, -5, 0.118, 3, 5, 1, 0}

## The roster method

One way to represent a set is with the roster or "tabular" form. With this method, we list all elements of our set, separate them with commas, and place them between braces.

For example:{3, 2, 4, 2, 1, 6, 3, 6, 7, 2}

This method is far more common than any other, and it's what we'll typically see when solving statistics and probability problems.

## The set-builder method

Another option is the set-builder method. This strategy involves establishing that all elements of the set share the same property.

For example: $Z=\left\{x:x\mathrm{integer}\right\}$

This essentially means: "The set Z equals all the values of x such that x is an integer."

Here''s another example of the set-builder method: $M=\left\{x|x\ge 3\right\}$

This essentially means: "All real numbers x such that x is greater than 3." For example, 3.1 is within this set while 2 is not.

## Empty sets

You might also get a set that has *no* elements whatsoever. For example, we might be measuring rainy days in the month of July. But what happens when it doesn't rain for a single day in July?

In this case, we would create a special set called an "empty set." Empty sets are very easy to represent since we can just use this symbol: "$\mathrm{\xe2\x88\x85}$."

## Elements of a set

If x is an element of a set A, we can write this in the following formula:

$x\in A$

If x is *not* an element of A, we can use this formula instead:

$x\notin A$

If we know that all values within set Z must be an integer, we can write something like this:

$\mathrm{-862}\in Z$

Why? Because -862 is an integer.

If set M includes all real numbers greater than 3, we can write this:

$M\notin 2.9$.

## Topics related to the Sets

Fundamental Counting Principle

## Flashcards covering the Sets

## Practice tests covering the Sets

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

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