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Introduction to Groups and Sets in Algebra

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  • 0:03 Intro to Groups & Sets
  • 2:17 Sets
  • 3:16 Numerical Sets
  • 4:17 Group Operations
  • 6:18 Group Rules
  • 10:18 Examples
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 20 years of experience teaching psychology and math related courses at the university level.

As an introduction to the algebraic concepts of sets and groups, this lesson covers the difference between the two concepts and how to determine if a set is a group. To complete this lesson, we'll take a look at some practice examples.

Groups and Sets: Definitions

Hello, and welcome to this lesson on groups and sets. Algebraic groups and sets have a very special relationship to each other. In fact, groups are always sets, but sets are not always groups. Confused?

Don't be. It is just like saying that all vowels are letters, but not all letters are vowels. See?

Maybe the definitions will help you see the relationship. A set is a collection of items called elements. You can have sets of any type of item (like cars, action figures or numbers).

A group is a set combined with an operation that follows four specific algebraic rules.

So, you see, a set on its own is not necessarily a group, but a set that is combined with an operation and follows the rules is a group.

Let's use people as a living example of the concepts for this lesson.

These people, collected together on the screen, are a set of people:

Set of people
image of multiple people

To be a group, maybe the group could be a club or any other organization, you would expect all the people you see to have a shared purpose in what they do together (that would be the operation - what they do), and you would expect to find some rules of behavior for everyone involved as well.

In this set of women, we have a group that I'll call the Crafters Club:

The Crafters Club is a group.
image with two groups of people

All of the women in the Crafters Group are women, obviously, but not all of the women in our original set are crafters. So, the women in the Crafters Group all share a specific operation - that of crafting. They probably have rules of behavior to abide by when they're in their group.

Sets

So, let's get a closer look at sets to really understand them before we go further with groups. Like I said, a set is just a collection of items that we call elements. Say you have a bunch of people. There are young boys and girls as well as adult men and women. But, you only want to talk to the adult women.

You might write a long note explaining who you'd like to speak to, or you could simply write the request in set notation, which is the proper algebraic format for indicating a set. You could write this: {adult | adult = woman}. This translates in English to 'I would like a set of all the adults, such that (which means as long as) the adult is a woman.'

See how much easier it was to use set notation?

Numerical Sets

Sets work the same with numbers. What if you wanted to indicate that your set included all of the positive integers?

Well, you could write down every single positive number in existence. Actually, you couldn't do that because the list would go on infinitely. You'd never get to the end. So, how do you do it?

Like this: {x is a member of the group integers | x > 0}. That's how you do it. This set notation translates into all numbers that are integers and greater than 0. If you are unsure of the translation here, or would like extra practice with sets, please watch the lesson on Set Builder Notation.

But, how do you know if a set is also a group?

Group Operations

Remember our first living example of the club? The women would not be considered a club or a group if they were not all working toward a similar goal defined by the group, right? Here, they are crafters, and the group operation is to craft.

The first requirement for a set of numbers to be a group is to have some operation being performed on the set. In math, we know there are four operations: addition, subtraction, multiplication and division. Right? Wrong!

There are really only two operations: addition and multiplication. In algebra, when we subtract, we really just add the opposite: 5 - 5 is really 5 + (-5). The same is true with division. When we divide, we really just multiply the opposite: 5 / 5 is the same as 5 x 1/5. (The inverse of 5 is 1/5). So, understanding there are only two operations available really cuts down on the restrictions for determining if a set is also a group. We only have two operations to test.

Now, what do I mean when I say that a set has an operation? It just means that we are going to choose whether we will add or multiply the elements in a set to see if the group rules apply. And we say that the set is over the operation. The set of integers over addition means that we have chosen the operation addition to use with all of the integers in our set.

Group Rules

I've said that a set must have an operation and follow rules to be considered a group. There are four specific rules that a set must follow.

In terms of our living example, you can think of this like the rules for the club members. As well as having a common goal or action (like the operation in our number sets), club groups would have rules that all the members know and must follow in order to stay a member of the group. This is actually the same for our algebraic groups.

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