Existence, Properties, and Official Definition of Inverse functions.

I. When does a given function  f  have an inverse map that is also a function, and what are the properties of the inverse map?

Given a function  f : A → B we define the inverse map of  f  to be the map  f ^-1: B → A that is defined by saying that  f ^-1(b) = a means  f (a) = b. The inverse map  f ^-1  may or may not be qualified to be a function.

Theorem that tells us when the inverse map is a function

Theorems 7.2.2 and 7.2.3 combined tell us that if a function  f : A → B is both one-to-one and onto, then the inverse map  f ^-1  will have these properties:

• The map  f ^-1 will be a function (by Theorem 7.2.2).
• The map  f ^-1 will be both one-to-one and onto (by Theorem 7.2.3).

Proof (discussed in class on Wed Mar 28, this proof is way cooler than the book's proofs)

1. Suppose that a function  f : A → B is both one-to-one and onto.
2. Then we have two quantified statements about  f  that we know are true:
   • because  f  is a function, we know that ∀ a ∈ A ( ∃ ! b ∈ B (  f (a) = b ))
   • because  f  is a both one-to-one and onto, we know that ∀ b ∈ B ( ∃ ! a ∈ A (  f (a) = b ))
3. But by definition of inverse map  f ^-1, the expression  f (a) = b means the same thing same as  f ^-1(b) = a.
4. Replacing the expression  f (a) = b with the expression  f ^-1(b) = a in the above statements, we get
   • ∀ a ∈ A ( ∃ ! b ∈ B (  f ^-1(b) = a)) This tells us that  f ^-1: B → A is one-to-one and onto.
   • ∀ b ∈ B ( ∃ ! a ∈ A (  f ^-1(b) = a)) This tells us that  f ^-1: B → A is a function.

II. Missing Theorem A: What if a function  f  has an inverse map  f ^-1 that is also known to be a function?

The book does not present this theorem, but it is a very useful theorem, so I present it here and call it Theorem A.

Theorem A: If a function  f : A → B has an inverse map  f ^-1: B → A that is also known to be a function, then  f  and  f ^-1 will have these properties:

• The function  f  will be both one-to-one and onto.
• The function  f ^-1 will be both one-to-one and onto

Proof (cool proof similar to my proof of Theorems 7.2.2 and 7.2.3)

1. Suppose that a function  f : A → B has an inverse map  f ^-1: B → A that is known to be a function.
2. Then we have two quantified statements that know are true about  f  and  f ^-1:
   • because  f  is a function, we know that ∀ a ∈ A ( ∃ ! b ∈ B (  f (a) = b ))
   • because  f ^-1 is a function, we know that ∀ b ∈ B ( ∃ ! a ∈ A (  f ^-1(b) = a ))
3. But by definition of inverse map  f ^-1, the expression  f (a) = b means the same thing same as  f ^-1(b) = a.
4. Replacing the expression  f (a) = b with the expression  f ^-1(b) = a in the first statement, and replacing the expression  f ^-1(b) = a with the expression  f (a) = b in the second statement, we get
   • ∀ a ∈ A ( ∃ ! b ∈ B (  f ^-1(b) = a)) This tells us that  f ^-1: B → A is one-to-one and onto.
   • ∀ b ∈ B ( ∃ ! a ∈ A (  f (a) = b)) This tells us that  f : A → B is one-to-one and onto.

III. Restating everything with the symbol g instead of the symbol  f ^-1

It is useful to restate all of our results so far with the symbol g instead of the symbol  f ^-1.

Reworded Definition of the Inverse Map

Given a function  f : A → B we define the inverse map g ^-1: B → A by saying that g(b) = a ⇔  f (a) = b. The inverse map g may or may not g might not be qualified to be a function.

Reworded Theorem that tells us when the inverse map is a function

Theorems 7.2.2 and 7.2.3 combined and reworded using the symbol g instead of the symbol  f ^-1

If a function  f : A → B is both one-to-one and onto and g: B → A is the inverse map (that, g is the map defined by g(b) = a ⇔  f (a) = b), then g will have these properties:

• The map g will be a function (by Theorem 7.2.2).
• The map g will be both one-to-one and onto (by Theorem 7.2.3).

Reworded Missing Theorem A

Missing Theorem A reworded using the symbol g instead of the symbol  f ^-1

Theorem A: If a function  f : A → B has an inverse map g: B → A that is also known to be a function, then  f  and g will have these properties:

• The function  f  will be both one-to-one and onto.
• The function g will be both one-to-one and onto

IV Defining the Inverse Function and Restating our Theorems with that Terminology

Inspired by our reworded Theorem 7.2.2 and Theorem 7.2.3 and Theorem A, we make the following definition.

With this new terminology, we can streamline the statements of Theorems 7.2.2 and Theorem 7.2.3, and Theorem A.

Theorem 7.2.2 and Theorem 7.2.3 tell us:

If a function  f  is both one-to-one and onto,
then  f  has an inverse function g, and g is also both one-to-one and onto.

Theorem A tells us:

If a function  f  has an inverse function g
then both  f  and g are one-to-one and onto.