# When does injectivity imply surjectivity?

I'm aware of the existence of this question: Surjectivity implies injectivity

However, the question is regarding a finite set $S$. I was wondering, though: What happens when $S$ is an infinite set? Zhen Lin addresses this cases in his answer, by saying that it ceases to be necessarily true, for example $f:\mathbb{N}\rightarrow \mathbb{N}$ defined by $x \mapsto x+1$ is injective but not surjective.

My question is: what happens if $S = \mathbb{R}$? Constructing a counterexample for $S=\mathbb{N}$ seems simple enough, but I'm struggling to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ that's injective but not surjective. Does such a function even exist? If so, how to construct it?

And perhaps a more general question (maybe too broad): For which infinite sets $S$ there is a function $f:S\rightarrow S$ such that $f$ is injective but not surjective?

• $\exp{}{}{}{}{}$ Jun 22, 2014 at 6:08
• That every infinite set is Dedekind-infinite requires some very weak form of choice. For example, the assumption that every infinite set contains a countable subset is sufficient.
– user123641
Jun 22, 2014 at 6:23
• If you already have such an $f$ for $\mathbb{N}$, just extend it to $\mathbb{R}$ by defining it as the identity on $\mathbb{R} \cap \mathbb{N}^C$ Jun 22, 2014 at 13:08
• This is, to some extent, related: Equivalent characterisations of Dedekind-finite proof Nov 24, 2016 at 10:14

The function $f(x) = \arctan x$ is injective but clearly not surjective on $\mathbb{R}$.
$f(x)=\arctan x$, for example, is injective but not surjective.
For your second question, let $S$ be an infinite set and $f:S\to S$ be any injection. Then fix some $x_0$ in $S$. Then, since $S$ is infinite, $|S|=|S\setminus\{x_0\}|$ ($|A|$ is the cardinal of a set $A$), so there exists a bijection $\phi$ of $S$ onto $S\setminus\{x_0\}$. Then $\phi\circ f$ seen as an application from $S$ into $S$ is injective but not surjective.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# When does injectivity imply surjectivity?

I'm aware of the existence of this question: Surjectivity implies injectivity

However, the question is regarding a finite set $S$. I was wondering, though: What happens when $S$ is an infinite set? Zhen Lin addresses this cases in his answer, by saying that it ceases to be necessarily true, for example $f:\mathbb{N}\rightarrow \mathbb{N}$ defined by $x \mapsto x+1$ is injective but not surjective.

My question is: what happens if $S = \mathbb{R}$? Constructing a counterexample for $S=\mathbb{N}$ seems simple enough, but I'm struggling to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ that's injective but not surjective. Does such a function even exist? If so, how to construct it?

And perhaps a more general question (maybe too broad): For which infinite sets $S$ there is a function $f:S\rightarrow S$ such that $f$ is injective but not surjective?

• $\exp{}{}{}{}{}$ Jun 22, 2014 at 6:08
• That every infinite set is Dedekind-infinite requires some very weak form of choice. For example, the assumption that every infinite set contains a countable subset is sufficient.
– user123641
Jun 22, 2014 at 6:23
• If you already have such an $f$ for $\mathbb{N}$, just extend it to $\mathbb{R}$ by defining it as the identity on $\mathbb{R} \cap \mathbb{N}^C$ Jun 22, 2014 at 13:08
• This is, to some extent, related: Equivalent characterisations of Dedekind-finite proof Nov 24, 2016 at 10:14

The function $f(x) = \arctan x$ is injective but clearly not surjective on $\mathbb{R}$.
$f(x)=\arctan x$, for example, is injective but not surjective.
For your second question, let $S$ be an infinite set and $f:S\to S$ be any injection. Then fix some $x_0$ in $S$. Then, since $S$ is infinite, $|S|=|S\setminus\{x_0\}|$ ($|A|$ is the cardinal of a set $A$), so there exists a bijection $\phi$ of $S$ onto $S\setminus\{x_0\}$. Then $\phi\circ f$ seen as an application from $S$ into $S$ is injective but not surjective.
user
The text passage is a discussion about the conditions under which injectivity implies surjectivity for functions, particularly focusing on the case when the domain is an infinite set, such as the real numbers. It includes a specific example of a function that is injective but not surjective and a more general construction for any infinite set.

TYPE: NO_QUESTION

However, the passage does contain several questions, but they are not mathematical questions in the sense of requiring a numerical or calculative answer. They are more in the form of inquiries about the existence of certain types of functions and the properties of infinite sets. Therefore, the correct classification should be:

TYPE: MULTIPLE_QUESTIONS