### One-to-One

##### 1. One-to-one and uniqueness

After the discussion about the relation between onto and existence,
we turn to uniqueness from the transformation viewpoint.
Again the relation can be seen through everyday life questions.

Example For the transformation

`Capital City`: country → city

we already know the city of Beijing is in the range. Now we ask

How many cities has Beijing as the capital city?

The answer is that Beijing is the capital city of a *unique* contry - China. In other words,

For **b** =Beijing, the solution to the equation
`Capital City`(**x**) = **b** is unique.

In fact, we know that the uniqueness for any capital city **b** in the world.
This means that for any city **b**, we have

`Capital City`(country_{1}) = `Capital City`(country_{2}) ⇒
country_{1} = country_{2}.

In other words, countries with the same capital must be the same country.

The discussion leads to the following definition.

A transformation **T**: **X** → **Y**
is one-to-one (injective) if
**T**(**x**) = **T**(`x'`)
implies **x** = `x'`.

We note that the condition is also equivalent to

**x** ≠ `x'` ⇒
**T**(**x**) ≠ **T**(`x'`)

(different elements are transformed to different elements)

If a transformation is not one-to-one, then it is several-to-one:
Different elements of `X` may have the same image.

More discussion on one-to-one transformations can be found here.

Example For the transformation

`Instructor`: course → professor

the following are equivalent statements

`Instructor` is one-to-one
`Instructor`(course_{1}) = `Instructor`(course_{2})
⇒ course_{1} = course_{2}
(courses taught by the same professor are the same course)
- course
_{1} ≠ course_{2} ⇒
`Instructor`(course_{1}) ≠ `Instructor`(course_{2})
(different courses are taught by different professors)
- No professor teaches more than one courses

The answer is (almost) definitely no.

Because there are millions of people at age 20, the transformation
`Age`: people → number is not one-to-one.

Example The transformation
`Square`: **R** → **R** is two-to-one at nonzero numbers (and is one-to-one at 0).
Overall, the transformation is not one-to-one.

For `Reflection in x-axis`, note that two points with the same reflections
must be the same point. Therefore the transformation is one-to-one.

Similarly, the `Rotation` transformation is also one-to-one.