# Prove that if g(f(x)) is injective then f is injective

• MHB
• cbarker1
In summary, the conversation discusses a proof that shows if g(f(x)) is injective, then f is also injective. The proof involves supposing that g(f(x)) is injective and using this to show that f is also injective. The other person provides a suggestion for further clarification.
cbarker1
Gold Member
MHB
Dear Everybody,
Question:
"Prove that if g(f(x)) is injective then f is injective"
Work:
Proof: Suppose g(f(x)) is injective. Then g(f(x1))=g(f(x2)) for some x1,x2 belongs to C implies that x1=x2. Let y1 and y2 belongs to C. Since g is a function, then y1=y2 implies that g(y1)=g(y2). Suppose that f(x1)=f(x2). Then g(f(x1))=g(f(x2)). Therefore f is injective. QED

You haven't proved $f$ is injective. To fix it, suppose $f(x_1) = f(x_2)$. Then $g(f(x_1)) = g(f(x_2))$. Injectivity of $g\circ f$ implies $x_1 = x_2$. Thus $f$ is injective.

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Great proof! Your explanation is clear and concise. One small suggestion I have is to define what C represents in the proof, just for clarity. Other than that, well done! Keep up the good work.

## 1. What does it mean for a function to be injective?

An injective function is one-to-one, meaning that each input has a unique output. In other words, no two different inputs can result in the same output.

## 2. Can you give an example of a function that is injective?

One example of an injective function is f(x) = x, where the input and output are the same value. For instance, f(2) = 2 and f(3) = 3, so no two inputs can have the same output.

## 3. What does it mean for g(f(x)) to be injective?

This means that the composition of two functions, g and f, results in an injective function. In other words, the output of f(x) is unique, and when that output is used as the input for g(x), the resulting output is also unique.

## 4. How does the injectivity of g(f(x)) relate to the injectivity of f(x)?

If g(f(x)) is injective, then f(x) must also be injective. This is because if the output of f(x) is unique, then when that output is used as the input for g(x), the resulting output must also be unique.

## 5. Can you prove that if g(f(x)) is injective, then f(x) is injective?

Yes, it can be proven using a direct proof. Assume that g(f(x)) is injective, and let a and b be two different inputs for f(x). Since f(x) is injective, f(a) is not equal to f(b). Then, when these outputs are used as inputs for g(x), g(f(a)) is not equal to g(f(b)). Therefore, f(x) must also be injective.

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