The Inverse Property

The Inverse Property:

A set has the inverse property under a particular operation if every element of the set has an inverse. An inverse of an element is another element in the set that, when combined on the right or the left through the operation, always gives the identity element as the result. Again, this definition will make more sense once we’ve seen a few examples.

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), y represents the a special element called the inverse of x (see definition below), e represents the identity element of the set, and the symbol # represents our operation, then the inverse property, for A with the operation # would be:

For any element x of the set, there is another element y of the set so that x#y = e and y#x = e.

(It is important to note that a set under an operation must have the identity PROPERTY before it can have the inverse property: if there is no identity, then the definition of the inverse doesn’t make sense! So if a set under an operation does not have the identity PROPERTY, then we know already that it does not have the inverse property!)

This means that the inverse property only holds if every single element in the set has an inverse!

Also notice that the inverse must be the same on the right and on the left! If we have two elements, y and z so that x#y = e but y#x ≠ e and z#x = e but x#z ≠ e, then we don’t have an inverse. Neither y nor z is an inverse, because they don’t give the identity as the result when they act on x both on the right and on the left.

Something else you should notice: if x is the inverse of y, then y is always the inverse of x. Inverses always come in pairs! (Although we can have an element be an inverse of itself – see below.)

One more important point: the identity element is always its own inverse. For example, if e is the identity element, then e#e=e. So by definition, when e acts on itself on the left or the right, it leaves itself unchanged and gives the identity element, itself, as the result!

Also note that while elements usually don’t have more than one inverse, it is possible for an element to have more than one inverse.

Let’s look at some examples so that we can understand the inverse property more clearly:

First let’s look at a few infinite sets with operations that are already familiar to us:

a) The set of integers has the inverse property under the operation of addition, because it has the identity element 0, and it is true that for any integer x, x+(−x)=0 and −x+x=0. So −x is the inverse for x in the set of integers under the operation of addition because it gives the identity 0 as the result whenever it acts on x from the right or the left.

 

b) The set of natural numbers does not have the inverse property under the operation of addition, because while −x would normally be the inverse for x under addition, there are no negative numbers in the set of natural numbers. The inverse must be in the set in order for the inverse property to hold!

 

c) The set of integers does not have the inverse property under the operation of division, because the set of integers under the operation of division does not have an identity element (we showed this in the last section)! So it cannot have inverse S because the definition of an inverse requires the existence of an identity element to make sense!

 

to see some more examples!

It can be a little bit more difficult, if all we have is an operation table, to tell whether or not a set has the inverse property under a particular operation. To figure out how to do this,

Let’s look at a few simple sets with operation TABLES and check to see if they have the inverse property: 

e) 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

We recall from the examples in the last section that the identity element of this set is a.

To find any inverses that might exist, first we look at the table and find any place where the result of an operation on two elements is the identity element a:

*	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

b*b= a

c*c= a

So every element has an inverse:

a is the inverse of a

b is the inverse of b

c is the inverse of c

In this set, every element is its own inverse!

So the set {a,b,c} under the operation * defined by the operation table above does have the inverse property!

 

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	b
c	b	c	a

We recall from the previous section that this table does not have an identity element. So we already know that this set does not have the inverse property!

 

g) Let’s consider the operation ^ acting on the set {β,γ,δ} given by the following operation table:

^	β	γ	δ
β	β	β	γ
γ	β	γ	δ
δ	δ	δ	β

We recall from the previous section that this set does have an identity element under the operation of ^.

So now let’s try to find any inverse S that might exist on the table:

 

To find any inverses that might exist, first we look at the table and find any place where the result of an operation on two elements is the identity element γ:

^	β	γ	δ
β	β	β	γ
γ	β	γ	δ
δ	δ	δ	β

From the table we can see that:

γ^γ= γ

So γ, because it is the identity, is it’s own inverse.

But there is no element x so that x^β=γ or β^x=γ, so β does not have an inverse!

And there is no element x so that x^δ=γ or δ^x=γ, so δ does not have an inverse!

 

We need every element to have an inverse in order for the set under the given operation to have the inverse property!

So the set {β,γ,δ} under the operation ^ defined by the operation table above does not have the inverse property!

 

h) Let’s consider the operation $ acting on the set {β,γ,δ,λ} given by the following operation table:

$	β	γ	δ	λ
β	λ	β	γ	β
γ	β	γ	δ	λ
δ	γ	δ	β	γ
λ	γ	λ	δ	δ

We can see that this set under the operation $ does have an identity element. The identity element is γ.

So now let’s try to find any inverse S that might exist on the table:

 

To find any inverses that might exist, first we look at the table and find any place where the result of an operation on two elements is the identity element γ:

$	β	γ	δ	λ
β	λ	β	γ	β
γ	β	γ	δ	λ
δ	γ	δ	β	γ
λ	γ	λ	δ	δ

From the table we can see that:

β$δ= γ

γ$γ= γ

δ$λ= γ

δ$β= γ

λ$β= γ

 

To find any inverse S that might exist, let’s break these down so that we can look at each element of the set one at a time:

First let’s consider γ:

γ$γ= γ

So γ, because it is the identity, is it’s own inverse.

 

Now let’s look at δ:

δ$λ= γ

β$δ= γ      δ$β= γ

So while δ$λ= γ , because λ$δ≠γ, λ is not an inverse of δ.

However, because β$δ= γ and δ$β= γ , β is an inverse of δ!

 

Next let’s consider β:

λ$β= γ

β$δ= γ δ$β= γ

So while λ$β= γ , because β$λ≠γ, λ is not an inverse of β.

However, because β$δ= γ and δ$β= γ , δ is an inverse of β!

 

And finally, let’s look at λ:

δ$λ= γ

λ$β= γ

So because δ≠β, there is no element x so that x^λ=γ and λ^x=γ. Therefore, λ does not have an inverse!

 

So γ, δ, and β have inverse S in this set under this operation, but λ does not have an inverse .

We need every element to have an inverse in order for the set under the given operation to have the inverse property!

So the set {β,γ,δ,λ} under the operation $ defined by the operation table above does not have the inverse property!

to see some more examples.

 

 

Now return to Blackboard to answer Group Lecture Questions 5: Inverse Property!