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

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.

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.
 Rule 1: The group must contain an identity. An identity is an element that leaves all elements of the set unchanged when combined with the given operation. While that sounds complicated, it is really as simple as zero for addition and one for multiplication because anything + 0 = the same thing and anything x 1 also = the same thing. See, no change though the operation has been performed.
 Rule 2: The group must have an inverse. An inverse is an opposite element that, when combined with the operation, will result in the group identity, that thing from Rule 1. Again, this sounds complicated, but isn't really. The identity for addition is zero, so an example of an inverse over addition would be 6 + 6 = 0. The identity in multiplication is 1, so an example of an inverse would be 1/2 x 2 = 1. I bet you already knew that, just didn't know they had actual official names.
 Rule 3: The operation must be associative (this means that the order of the elements during the operation does not matter). For example: (1 + 2) + 3 is the same as 1 + (2 + 3) and (2 x 3) x 4 = 2 x (3 x 4). In these examples, the order of operations would not alter the end result. You can imagine that if the operations were not addition or multiplication, performing the operations in a different order would result in different answers. If you aren't quite sure why, please review the lesson covering Order of Operations.
 Rule 4: The group must be closed. A closed group means that the result of an operation performed on any elements of the group is also an element of the group. Now, this one seems really convoluted, doesn't it? How could the result NOT be a part of the group? Consider the set S = {1, 2, 3}. Let's look first at addition: 1 + 2 = 3; that's fine. 1 + 3 = 4; Oh, no! 4 is NOT in our set. This is not a group. The same is true for multiplication with this set: 1 x 2 = 2, 1 x 3 = 3. So far, so good. But, 2 x 3 = 6; 6 is not in the set, so the rule is broken.
All rules must apply for a set to be a group! If even one rule is broken, then the set is not a group.
Examples
So, do you think you have it? Let's try.
Is the set of integers a group over addition?
Well, we have an operation defined so we can move on to the rules.
 Rule 1: Identity. Yes, the identity for addition is 0, and it is a member of the set of integers.
 Rule 2: Inverse. Yes again. All integers have an additive inverse, which are members of the set of integers. 1 and 1 are examples of this additive inverse.
 Rule 3: Associative. Yes again. Within addition of integers, the order that integers are added does not change the end result.
And, finally…
 Rule 4: Closed. Yes again. When any integer is added to any other integer, the result is always an integer.
So, yes, the set of integers is a group over addition!
What about multiplication? Is the set of integers a group over multiplication?
We can evaluate this the same way:
 Rule 1: Identity. Yes, the multiplicative identity for integers is 1, and 1 is an integer.
 Rule 2: Inverse. No! The multiplicative inverse of the integer 2 is ½, and ½ is NOT an integer; thus this rule does not apply.
We can stop there. One rule does not apply, so the set is not a group. No, the set of integers is not a group over multiplication!
For any set, just apply the rules using the chosen operation, and you will see quickly whether or not the set is a group.
Lesson Summary
This lesson has been packed with quite a lot of information. Let's review the highlights using our living example to help. First, a set is just a collection of elements, like our people here.
We could have a set of just the women if we wanted. A set might have a condition, (like only the women), but there are no other rules that apply. In algebra, a set is noted using brackets and defining any condition, like this: {x  x >0 }
A group is a special type of set that involves an operation performed on the set and follows four rules. This is similar to the idea of a club, like our Crafters Group. The operation is the shared goal of crafting, and the rules are whatever rules the group has for its members.
In algebra, the operations are addition and multiplication, and the rules are identity, inverse, associative and closed.
I hope this lesson has helped you become more familiar with the concepts of sets and groups in algebra. Thanks for watching. Bye!
Learning Outcomes
Following this lesson, you'll be able to:
 Define set and group
 Identify the notation for a set
 List the four rules that a set must meet in order to be considered a group
 Explain why there are only two types of operations in algebra