One-To-One Functions

Let f: A B, a function from a set A to a set B. f is called a one-to-one function or injection, if, and only if, for all elements a1 and a2 in A, if f(a1) = f(a2), then a1 = a2 Equivalently, if a1  a2, then f(a1) f(a2).

Conversely, a function f: A B is not a one-to-one function elements a₁ and a₂ in A such that f(a₁) = f(a₂) and a₁ a₂.

In terms of arrow diagrams, a one-to-one function takes distinct points of the domain to distinct points of the co-domain. A function is not a one-to-one function if at least two points of the domain are taken to the same point of the co-domain. Consider the following diagrams:

                                

    A one-to-one function                                     A function that is not one-to-one

One-To-One Functions on Infinite Sets

To prove a function is one-to-one, the method of direct proof is generally used. Consider the example:

Example: Define f : R R by the rule

f(x) = 5x - 2 for all x R

Prove that f is one-to-one.

Proof: Suppose x₁ and x₂ are real numbers such that f(x₁) = f(x₂). (We need to show x₁ = x₂ .)

5x₁ - 2 = 5x₂ - 2

Adding 2 to both sides gives

5x₁ = 5x₂

Dividing by 5 on both sides gives

x₁ = x₂

We have proven that f is one-to-one.

On the other hand, to prove a function that is not one-to-one, a counter example has to be given.

Example: Define h: R R is defined by the rule h(n) = 2n². Prove that h is not one-to-one by giving a counter example.

                 Counter example:

                 Let n₁ = 3 and n₂ = -3. Then

                                  h(n₁) = h(3) = 2 * 3² = 18 and

                                 h(n₂) = h(-3) = 2 * (-3)² = 18

                 Hence h(n₁) = h(n₂) but n₁  n₂, and therefore h is not one-to-one.

Onto Functions

Let f: AB be a function from a set A to a set B.  f is called onto or surjective if, and only if, all elements in B can find some elements in A with the property that y = f(x), where y B and x A. f is onto y   B, x  A such that f(x) = y.

Conversely, a function f: A B is not onto y in B such that x  A,  f(x) y.

In arrow diagram representations, a function is onto if each element of the co-domain has an arrow pointing to it from some element of the domain. A function is not onto if some element of the co-domain has no arrow pointing to it. Consider the following diagrams:

                      An onto function                                      A function that is not onto

Proving or Disproving That Functions Are Onto

Example: Define f : R R by the rule f(x) = 5x - 2 for all xR. Prove that f is onto.

Proof: Let y R. (We need to show that x in R such that f(x) = y.)

If such a real number x exists, then 5x -2 = y and x = (y + 2)/5. x is a real number since sums and quotients (except for division by 0) of real numbers are real numbers. It follows that

f(x) = 5((y + 2)/5) -2   by the substitution and the definition of f

       = y + 2 -2

       = y                        by basic algebra

Hence, f is onto.

Example: Define g: Z Z by the rule g(n) = 2n - 1 for all n Z. Prove that g is not onto by giving a counter example.

Counter example:

The co-domain of g is Z by the definition of g and 0 Z. However, g(n) 0 for any integer n.

If g(n) = 0, then

2n -1 = 0

2n  = 1       by adding 1 on both sides

n  = 1/2      by dividing 2 on both sides

But 1/2 is not an integer. Hence there is no integer n for g(n) = 0 and so g is not onto.

One-To-One Correspondences

A function f : A B can be both one-to-one and onto at the same time. This means that given any element a in A, there is a unique corresponding element b = f(a) in B. Also given any element b in B, there is an element a in A such that f(a) = b as f is onto and there is only one such b as f is one-to-one. In this case, the function f sets up a pairing between elements of A and elements of B that pairs each element of A with exactly one element of B and each element of B with exactly one element of A. This pairing is called one-to-one correspondence or bijection. When depicted by arrow diagrams, it is illustrated as below:

Inverse Functions

If there is a function f which has a one-to-one correspondence from a set A to a set B, then there is a function from B to A that "undoes" the action of f. This function is called the inverse function for f.

Suppose f: A B is a one-to-one correspondence (f is one-to-one and onto). Then there is a function f -1: B A. It is defined as follows: f -1(b) = a   b = f(a) f -1 is the inverse function of f.

The diagram below shows the fact that an inverse function sends each element back to where it came from.

Finding an inverse function for a function given by a formula:

Example: Define f: R R by the rule f(x) = 5x - 2 for all x R. It has been already shown above that f is one-to-one and onto. Hence f is a one-to-one correspondence and has an inverse function f ^-1.

Solution: By the definition of f ^-1,

f ^-1(y) = x    such that    f(x) = y

But ; f(x) = y

 5x-2 = y                by the definition of f

 x   = (y + 2)/5        by basic algebra

Hence f ^-1(y) = (y + 2)/5.

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