# Linear Algebra Proof on Composition of One-to-One Functions

• jrk012
In summary, to prove that the composition of one-to-one functions is also a one-to-one function, we can use a proof by contradiction. Assume that f(g(x)) is not one-to-one, then there exist a, b in set A such that f(g(a)) = f(g(b)) and a ≠ b. However, since f and g are one-to-one functions, there must exist a unique x in set B such that f(x) = f(g(a)) = f(g(b)). This contradicts the assumption that f(g(x)) is not one-to-one, thus proving that the composition of one-to-one functions is also a one-to-one function.
jrk012

## Homework Statement

Prove that the composition of one-to-one functions is also a one-to-one function.

## Homework Equations

A function is one-to-one if f(x1)=f(x2) implies x1=x2. Composition is (f*g)(x)=f(g(x)). Proof-based question.

## The Attempt at a Solution

A one-to-one function does not repeat the image. If we have two one-to-one function f(x) and g(x), then f and g do not repeat their images. Then, when then the composition, for example f(g(x)), for all x, g(x) does not repeat the image and after that applying f(x) also does not repeat the image, therefore the composition of the function is one-to-one as well.

Is this a good proof for the question?

I would suggest a proof by contradiction. Let g be a one-to-one function from set A to set B, f a one-to-one function from set B to set C. Suppose the statement were not true- that f(g(x)) is not one-to-one. Then there exist a, b, a not equal to b, in A such that f(g(a))= f(g(b)). Since f is one-to-one, there must exist a unique x in b such that f(x)= f(g(a))= f(g(b)). Can you complete this?

If in doubt, proof by contradiction!

## 1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of vector operations, matrices, and linear transformations.

## 2. What is a One-to-One Function?

A one-to-one function is a type of function in which each element in the domain maps to a unique element in the range. This means that no two elements in the domain can map to the same element in the range.

## 3. What is the Composition of One-to-One Functions?

The composition of one-to-one functions is a mathematical operation that combines two one-to-one functions to create a new function. The output of one function becomes the input of the other function, resulting in a new function that is also one-to-one.

## 4. How do you prove that the composition of two One-to-One Functions is also One-to-One?

To prove that the composition of two one-to-one functions is also one-to-one, we need to show that if f and g are one-to-one functions, then the composition of f and g, denoted as f(g(x)), is also one-to-one. This can be proven by showing that for any two distinct elements in the domain of f(g(x)), their outputs will also be distinct.

## 5. How does Linear Algebra relate to the Composition of One-to-One Functions?

Linear Algebra plays a crucial role in understanding the composition of one-to-one functions. It provides the tools and concepts necessary to prove the properties of one-to-one functions and their compositions. The concepts of linear transformations and matrix operations are particularly useful in this context.

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