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### Suppose that f is a function from A to B, where A and B are finite sets |A| = |B|. Show that f is one-to-one if and only if it is onto. [duplicate]

Suppose that f is a function from A to B, where A and B are finite sets |A| = |B|. Show that f is one-to-one if and only if it is onto. Lets say we were to translate this statement using variables. ...
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### Proving a function is bijection [duplicate]

My problem is: Let $A$ and $B$ finite sets of equal cardinality. Show that if a function $f : A \to B$ is injective then it is also surjective. Similarly show that if $f$ is surjective, then it is ...
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### Cardinalities of the two sets and the condition of the function in between [duplicate]

So if I have two sets A and B and if |A| = |B| and if function f is a function from A to B and is injective, how can I prove that it is also surjective? Forgot to mention. A and B are finite sets.
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### Surjectivity implies injectivity

Let S be a finite set.Let F be a surjective function from S to S. How do I prove that it is injective?
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### Empty functions are not injective?

Many sources say that empty functions such as $f:\emptyset \rightarrow S$ are injective because it is a vacuous truth. But currently I am reading a book on axiomatic set by Patrick Suppes, and he ...
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### Counting the number of injective and surjective functions from a set to itself

I have a question about counting the number of injective (one-to-one) and surjective (onto) functions from $\{1,2,...,n\}$ to itself that satisfy $|f(i)-i| \leq 1, \forall i\in \{1,2,...,n\}$. I ...
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### Showing that if $f$ is surjective, then $m\geq n$ holds (where $m$ and $n$ are the number of elements in the domain and codomain respectively)

Let $X$ and $Y$ be sets with $m$ and $n$ elements respectively and let $f:X\rightarrow Y$ be a function from $X$ to $Y$. Show that if $f$ is surjective, then $m\geq n$. I know this should be ...
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### Non-bijective endomorphisms of finite groups

How can I prove that there are no finite groups $G$ where there exists an endomorphism $G\rightarrow G$ that is injective but not surjective, or other way round?
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### Prove a map X → X must be bijective

Let X be a finite set. Does there exist a map $α : X → X$ such that $α$ is surjective, but not injective? $$\text{No.}$$ \begin{align} \text{Surjective: }& \ \ {\displaystyle \forall y\in Y,\...
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### How to write a rigorous proof for this statement?

Prove that for finite set $X$, the function $f:X \to X$ is surjective if and only if it is injective I have the idea of proof in my mind but find it difficult to translate it into mathematical ...
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### is it true that an injective unary operation on a finite set is surjective?

To prove the statement:Any finite integral domain $R$ is a field. Let $r \in R$ , we define a function$f: R \rightarrow R$ by $f(x)=rx$. The proof says: this map is injective(i understand this), then ...
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