Basic Mathematics For Quantum Mechanics - Maps

What is a map?

A map is a way of relating items in one set to items in another set. If we have a map called f from set A to set B we write:

\[f : A \rightarrow B\]

Let’s see an example. If we have the sets

\[A = \{1, 2, 3\}\] \[B = \{4, 5, 6\}\]

Then we can define the map

\[f : A \rightarrow B\] \[f(x) = x + 3\]

So

\[f(2) = 2 + 3 = 5\]

Domains, Codomain and Range

The domain of a map is the set the map goes from and the codomain is the set it goes to. So if we have

\[f : A \rightarrow B\]

Then A is the domain and B is the codomain

The range of a function is the set of all the outputs of a function. This sounds very similar to the codomain but it is slightly different. To see this let’s look at an example. If we have the following map:

\[g : \mathbb{Z} \rightarrow \mathbb{Z}\] \[g(x) = x^2\]

Then the domain and the codomain are the set of integers. However, any integer squared is always positive so the range of the function is just the positive integers. So -1 is in the codomain but not in the range.

Injective maps

Some maps are called ‘injective’ or ‘one-to-one’. An injective map is one where no two inputs have the same output. Consider:

\[f : \{1, 2, 3\} \rightarrow \{12, 15, 16\}\] \[f(1) = 12\] \[f(2) = 16\] \[f(3) = 15\]

This is an injective map because every input goes to a different output. But

\[f : \{1, 2, 3\} \rightarrow \{12, 15, 16\}\] \[f(1) = 12\] \[f(2) = 16\] \[f(3) = 16\]

is not injective because 2 and 3 both go the same output.

When a map is injective, every output (e.g. everything in the range) has a unique inverse - we can tell exactly what the input was just from looking at the output.

Surjective maps

Surjective maps are also called ‘onto’ maps. A surjective map is one where the co-domain is the same as the range - this means the function produces every possible output.

For example:

\[g : \mathbb{Z} \rightarrow \mathbb{Z}\] \[g(x) = x^2\]

is not surjective as it will never produce -1.

But

For example:

\[h : \mathbb{Z} \rightarrow \mathbb{Z}\] \[h(x) = h + 3\]

is surjective as it will produce all the possible outputs e.g. all of the integers.

Bijective maps

Some maps are both injective and surjective so we call these maps bijective. Sometimes you’ll hear these maps called a bijection.

Binary operations

Some maps can take more than one input. For example, addition is an example of a map that takes two inputs and gives us back a single output.

A map f from two sets A and B to a third C is written like this:

\[f : A \times B \rightarrow C\]

For example, we might have the map

\[f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\] \[f(x, y) = x + y + 2\] \[f(3, 5) = 3 + 5 + 2 = 10\]

A map can take any number of inputs but if it takes two inputs we call it a binary operation and are very common.