Operations on Sets
Why are we interested in looking at operations on sets?
We have now learned all about sets, but one of the main reasons that set theory was originally invented by mathematicians was so that we could talk about ways in which the elements of the set might interact under a particular operation. In other words, sets can be important becuause their elements interact in interesting ways.
For example, one important feature of a set with a particular operation may be whether or not certain kinds of equations can always be systematically solved within that set. For example, you learned in previous algebra classes that linear equations can always be solved if we are working with real numbers. Let's look again at how you would normally solve a linear equation in the real numbers:
Consider x+2.5=8.2. Let's look at each of the steps we would need to take to solve this equation for x. We are going to write out EVERY SINGLE necessary step, even though you probably take a lot of these steps for granted when you solve a problem like this:
x + 2.5 = 8.2 |
We begin with this equation. |
(x + 2.5) + −2.5 = 8.2 + −2.5 |
In this step we add something to both sides which we know will "cancel out" the 2.5 which we want to get rid of on the left side. We add, essentially, a kind of "opposite" of 2.5. |
x + (2.5 + −2.5) = 5.7 |
In this step, we use the fact that (x + 2.5) + −2.5 = x + (2.5 + −2.5) to rewrite the left side of the equation. In other words, we know that when we are adding 3 numbers together, it doesn't matter how we group them; we can add the first two numbers together first, or we can add the last two numbers together first, and we will still get the same answer either way. |
x + 0 = 5.7 |
In this step, we simply compute that 2.5 + −2.5 = 0. |
|
x = 5.7 |
In this step, we use the fact that anything added to 0 is just itself. so, x + 0 = x. |
Notice that there are several important properties here that are essential to solving these kinds of equations:
-
We need to have something that acts as an "opposite" for any number that we incounter; otherwise, we won't be able to "cancel" things which we want to get rid of.
-
We need to know that when we have 3 things next to each other, we can perform the operation on the first to elements first or on the last two elements first, and that this will not change our answer.
-
We need to know that whenever we compute something, we will get an answer, and that answer will be in our set.
-
We need to know that when we put "opposites" together, we will get something that does not change an element when it is put together with another element. In other words, we need something that acts like 0 does under addition.
Also notice that the set we are working in will determine whether or not we have all of these four properties. For example, if you are only working in the set of whole numbers, then the equation x + 5 = 2 cannot be solved. It cannot be solved, because there is no solution to this equation which is in the set of whole numbers (the only solution is the number −3, which is not a whole number!). Also, notice that there is no "opposite" of 5 in the set of whole numbers - if we can only work with whole numbers, we cannot cancel out 5 here.
Because these four properties seem to be so important for solving equations, mathematicians gave a special name to any set which follows these properties under a given operation: any set under an operation which has these four properties is called a group.
In fact, there is a whole branch of modern mathematics research called group theory. Groups are sometimes studied because they represent an important physical phenomenon: for example, the set of all the moves you can make to a shape like a square or a triangle or a cube without distorting it will form a group. Groups may also be studied because they have certain practical applications: they are used as a tool to analyze the way proteins have acted on DNA and as a basis for some cryptographic systems, for example. However, groups may also be studied as purely abstract structures, without focusing on any of the physical phenomena which could be used to describe the group, because this kind of abstract study allows us to see identical structures and similar behaviors across seemingly unrelated fields. For example, Thompson's group, one of the groups on which I am doing research, can be explained in several different ways: it can be represented by a certain set of graphs made up of linear pieces which map the numbers between 0 and 1 on the x-axis to the numbers between 0 and 1 on the y-axis, but it can also be represented by the set of all the rules you could make up about how an operation could act on a string of numbers when the numbers are grouped differently. Mathematicians discovered these groups separately in totally different areas, and only after the abstract structure of Thompson's group was developed did mathematicians realize these two groups were in fact the "same". (If you found any of the content of this paragraph confusing, don't worry - you won't be tested in this - this is just here to give you a sense of why mathematicians are interested in groups in the first place).
So this section of the course is going to be based on developing the ideas we have just talked about in a more formal way so that we can identify whether or not certain sets with an operation are groups. We will begin by defining exactly what we mean when we talk about an operation.
What is an Operation?
An operation is just a word used to describe any procedure in which we take two elements of a set and follow some rule or procedure that gives us one element as the result.
To understand this new idea, it is important to see that an operation usually doesnt make sense unless we pair it with a SET that it can act on. In order to make sense of what an operation does, we have to pair the operation with a particular set so that we can see how it acts on the elements of the set.
For example, here are a few operations that you are already familiar with:
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Addition is an operation that can act on any set of numbers. For example, addition is an operation over the set of integers. If you take any two integers, you can add them and get a third integer as the result.
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Multiplication is an operation that can act on any set of numbers. For example, multiplication is an operation over the set of real numbers. If you take any two real numbers, you can multiply them and get a third real number as the result.
Notice that we always talk about an operation over a particular set.
Operation Tables
To clearly show how an operation acts on any two pairs of elements in a set, we can construct an operation table . An operation table for addition or multiplication is exactly like the addition tables or times tables that you might have studied in elementary school.
Not all operations will be familiar ones that youve already seen, though. We can make up any random operation on a set by just making a table. By making a table, we can show what result the operation gives for any set of elements.
For example:
We can make an addition operation table over the set of natural numbers this way:
+ |
1 |
2 |
3 |
4 |
5 |
|
1 |
2 |
3 |
4 |
5 |
6 |
|
2 |
3 |
4 |
5 |
6 |
7 |
|
3 |
4 |
5 |
6 |
7 |
8 |
|
4 |
5 |
6 |
7 |
8 |
9 |
|
5 |
6 |
7 |
8 |
9 |
10 |
|
. . . |
. . . |
. . . |
. . . |
. . . |
. . . |
|
The first row of this table tells us that 1+1=2 and 1+2=3 and 1+3=4, etc.
The second row of this table tells us that 2+1=3 and 2+2=4 ant 2+3=5, etc.
Notice that this table goes on forever! This is because the natural numbers go on forever.
if you want to see how an operation table would look for the set of
integers with the operation of multiplication!
The previous examples involved an infinite set: the natural numbers. But not all operation tables must involve operations on infinite sets.
For example:
1) Lets look at the operation of subtraction over the set {-4, 1, 2.5, 6}.
The table would look like this:
|
-4 |
1 |
2.5 |
6 |
-4 |
0 |
-5 |
-6.5 |
-10 |
1 |
5 |
0 |
-1.5 |
-5 |
2.5 |
6.5 |
1.5 |
0 |
-3.5 |
6 |
10 |
5 |
3.5 |
0 |
This first row of this table tells us that 4 (4) = 0 and 4 1 = 5 and 4 2.5 = 6.5, etc.
The second row of this table tells us that 1 (4) = 5 and 1 1 = 0 and 1 2.5 = 1.5, etc.
And so on for the other rows .
Notice that the first element in the operation must be taken from the first column and the second element in the operation must be taken from the first row.
to see another
example of an operation table for a finite
set.
By making up an operation table, we can make up an operation for a set that yields any result we like from two given elements!
So an operation does not have to act on a set of numbers! It could act on a set of letters, or words, or symbols, etc.!
Making up Our Own Operation Table:
Let’s make up an operation table that acts on the set {a,b,c}.
We will let the symbol * stand for this new operation on the set {a,b,c}.
Let’s say that we want the operation * to act on the elements of {a,b,c} in the following way:
a*a=a (This means that when we take the element a as the first element and the element a as the second element, then the operation * by definition gives us the element a as the result.)
a*b=b (This means that when we take the element a as the first element and the element b as the second element, then the operation * by definition gives us the element b as the result.)
a*c=c (This means that when we take the element a as the first element and the element c as the second element, then the operation * by definition gives us the element c as the result.)
b*a=b
b*b=a
b*c=c
c*a=c
c*b=c
c*c=a
We can put all of these rules for the operation * into an operation table, and get the following:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
c |
c |
c |
c |
a |
Reading Operation Tables
How do we read an operation table?
Let’s consider the operation table below:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
For the table above:
1) If we want to find out what the result of c*b is we would look at the table and reason this way:
Because c comes first, we would look at the row that begins with c:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
Because b is the second element, we would look at the column that begins with b:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
The element that is in the c row and the b column is then the answer to the expression c*b. The element c is in the c row and the b column so it is the result of the operation * acting on the elements c (as the first element) and b (as the second element).
So c*b=c.
2) If we want to find out what the result of b*c is we would look at the table and reason this way:
Because b comes first, we would look at the row that begins with b:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
Because c is the second element, we would look at the column that begins with c :
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
The element that is in the b row and the c column is then the answer to the expression b*c. The element b is in the b row and the c column so it is the result of the operation * acting on the elements b (as the first element) and c (as the second element).
So b*c=b.
Notice that c*b ≠ b*c! Be careful! The order in which you write the elements in an operation can change the result!
to a few more examples showing you how to read an operation table.
So for any operation table, the first column and the top row represent the two elements that are being acted on by the operation and all the other elements in the table represent the results obtained by letting the operation act on a pair of elements in the set. In the operation table below, the orange shaded part represents the area where all the results are.
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
Now return to Blackboard to answer Group Lecture Questions 1: Reading an Operation Table!