A set has the identity property under a particular operation if there is an element of the set that leaves every other element of the set unchanged under the given operation.
More formally, if x is a variable that represents any arbitrary element in the set we are looking at (let’s call the set we are looking at A), and the symbol # represents our operation, then the identity property, for A with the operation # would be:
There is some particular element of the set A called the identity element (which will denote by the letter e), so that x#e = x and e#x = x for any element of A that we plug in for the variable x. (It is important to understand here that e represents a specific fixed element of the set A, but x represents a variable that can change and take on the values of any element of the set A.)
This means that the identity property only holds for the set A and the operation # if, no matter what elements we take from A and put in place of x, x#e always has the result x and e#x always has the result x. This element e must be the same element for every different element we put in for x.
You should also be aware that it is only possible to have one identity element for each set. So if we’ve already found one identity element, we can stop. There won’t be another one to find in that set under that particular operation.
It is much easier to understand a property by looking at examples than it is by simply talking about it in an abstract way, so let's move on to looking at examples so that you can see exactly what we are talking about when we say that a set has the identity property:
First let’s look at a few infinite sets with operations that are already familiar to us:
a) The set of whole numbers has an identity element under the operation of addition, because it is true that for any whole number x, 0+x=x and x+0=x. So 0 is the identity element for the whole numbers under the operation of addition because it does not change any whole number when it is added to it.
b) The set of integers does not have an identity element under the operation of division, because there is no integer e such that x ÷ e = x and e ÷ x = x. It is true that x ÷ 1 = x for any x, but then 1 ÷ x ≠ x! In fact, the only thing we could put in for e that would make sure e ÷ x = x is x^{2}. But then our e would change for each value of x.
For example, if e=x^{2}:
If x=1, then e=1, but if x=2, then e=4.
So e would not be the same for every single element of the set of integers!
So we don’t have an identity element for the set of integers under the operation of division!
to see some more examples!
If all we have is an operation table, it can be a little bit more difficult to tell whether or not a set has the identity property under a particular operation. To figure out if there is a simple way to tell if a set with an operation table has the identity property under the given operation,
f) Here is an operation table for the set {a,b,c} and the operation *:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
c |
c |
c |
c |
a |
From the table we can see that:
a*a=a a*a=a
a*b=b b*a=b
a*c=c c*a=c
So the element a must be our identity element because a*x=x and x*a=x for every element of the set {a,b,c} that we put in for x! The element a doesn’t change any element that it operates on with the operation *! So the set {a,b,c} under the operation * defined by the operation table above does have the identity property!
(Notice that we must have e*e=e in order for e to be the identity element. The equations e*x=x and x*e=x must be true when we plug in e for x, because e is one of the elements of the set we are looking at, and every single element of the set must make the equations true!)
g) Here is an operation table for the set {a,b,c} and the operation ~:
~ |
a |
b |
c |
a |
c |
c |
b |
b |
c |
a |
a |
c |
b |
a |
b |
Remember, in order to be the identity element, an element must leave every single element in the set {a,b,c} unchanged both when it operates on it from the left and when it operates on it from the right!
a~a=c So a~a≠a! a~a=c So a~a≠a!
a~b=c So a~b≠b! b~a=c So b~a≠b!
a~c=b So a~c≠c! c~a=b So c~a≠c!
So the element a cannot be the identity element, because it changes every single element when it acts on it with the operation~!
b~a=c So b~a≠a! a~b=c So a~b≠a!
b~b=a So b~b≠b! b~b=a So b~b≠b!
b~c=a So b~c≠c! c~b=a So c~b≠c!
So the element b cannot be the identity element, because it changes every single element when it acts on it with the operation~!
c~a=b So c~a≠a! a~c=b So a~c≠a!
c~b=a So c~b≠b! b~c=a So b~c≠b!
c~c=b So c~c≠c! c~c=b So c~c≠c!
So the element c cannot be the identity element, because it changes every single element when it acts on it with the operation~!
So there is no identity element for this set {a,b,c} under the operation ~ represented by the table above!
Do we notice a pattern here in the tables when a set has an identity element under a certain operation? If you look at the tables, you’ll see that the identity element, if it exists, has rows and columns that just repeat the elements in the set as they are listed in the headings of the table!
1) The set {a,b,c} with the operation * as defined by this table had the identity element a:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
c |
c |
c |
c |
a |
Notice that the row that begins with a just repeats the elements in the header row at the top of the table: the header row has elements a, b, c, in that order, and the row that begins with a has a row with elements a, b, c, in that order!
But be careful! This is not enough yet to assert that a is the identity element! We have to look at the column that begins with a before we can make a decision!
So let’s look at the column that begins with a:
* |
a |
b |
c |
a |
a |
b |
c |
b |
b |
a |
b |
c |
c |
c |
a |
Notice that the column that begins with a just repeats the elements in the header column at the far left of the table: the header column has elements a, b, c, in that order, and the column that begins with a has elements a, b, c, in that order!
Because both the column and the row that begin with a look exactly like the column/row headers, we can say that a is the identity element.
2) The set {a,b,c} with the operation ~ as defined by this table did not have an identity element:
~
a
b
c
a
a
b
c
b
b
a
c
c
b
c
a
To see this, begin by looking at each row. Does one of the rows repeat the header row? In this case, the answer is yes. Only one row repeats the header row. The row that begins with a repeats the header row:
~
a
b
c
a
a
b
c
b
b
a
c
c
b
c
a
But be careful! This is not enough yet to assert that a is the identity element! We have to look at the column that begins with a before we can make a decision!
So let’s look at the column that begins with a:
~
a
b
c
a
a
b
c
b
b
a
b
c
b
c
a
This column almost repeats the header column, but notice that in the last cell of the column, we have a b instead of a c ! So the column that begins with a does not repeat the column header! So a is not the identity element. Because no other rows repeat the row header, we know that no other elements of the set {a,b,c} could be the identity element. So the set {a,b,c} under the operation ~ as described by the table above does not have an identity element!
h) Let’s consider the operation @ acting on the set {β,γ,δ}given by the following operation table:
@ |
β |
γ |
δ |
β |
β |
δ |
δ |
γ |
δ |
β |
γ |
δ |
γ |
δ |
β |
This set does not have an identity element under the operation of @ because the operation table does not have a single row that repeats the header row!
to see another example.
Now return to Blackboard to answer Group Lecture Questions 4: Identity Property!