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The text contains statements about the relationship between injective (one-to-one) and surjective (onto) functions. It presents claims and asks for evaluation or clarification of these claims. There's no specific math problem to solve numerically. Instead, it's about understanding the properties of injective and surjective functions.

AnalysisFunctionsInjectiveSurjectiveOne-to-oneOntoMappingSet Theory
2025/5/17

1. Problem Description

The text contains statements about the relationship between injective (one-to-one) and surjective (onto) functions. It presents claims and asks for evaluation or clarification of these claims. There's no specific math problem to solve numerically. Instead, it's about understanding the properties of injective and surjective functions.

2. Solution Steps

We will analyze the statements provided in the image. Injective functions map distinct elements of the domain to distinct elements of the codomain. Surjective functions map the domain onto the entire codomain (every element in the codomain has at least one corresponding element in the domain).
Statement 1: "Not all injective are surjective but all surjective are injective"
This statement is partially correct. "Not all injective are surjective" is true. Consider the function f:R−>Rf: R -> R defined as f(x)=exf(x) = e^x. This is injective because ex=eye^x = e^y implies x=yx=y. However, it is not surjective because the range of ff is (0,∞)(0, \infty) which is not the whole of RR. "All surjective are injective" is false. Consider the function f:R−>Rf: R -> R where f(x)=x3−xf(x) = x^3 - x. For every y∈Ry \in R, there exist some x∈Rx \in R such that f(x)=yf(x) = y. This function is surjective but not injective. For instance f(0)=f(1)=0f(0) = f(1) = 0
Statement 2: "Not all surjective are injective, all injective are some surjective"
The first part, "Not all surjective are injective," is true, as seen from our previous analysis. "All injective are some surjective" is not correct. A function is either surjective or not. Saying it is "some surjective" doesn't make sense. A better way to phrase a correct statement is that "some injective functions are surjective".
Statement 3: "All injective mapping are subjective but not all surjective are injective"
The first part, "All injective mapping are subjective" is false. Injective mappings are not always surjective. As stated before "Not all surjective are injective" is true.

3. Final Answer

The statements in the image attempt to describe the relationship between injective and surjective functions, but they contain inaccuracies. Here is a summary of what is true and what needs correction:
* Not all injective functions are surjective.
* Not all surjective functions are injective.
* Some injective functions are surjective.
* Some surjective functions are injective.

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