Give an example of two functions $\alpha,\beta$ on a set $A$ such that $\alpha\circ\beta=\mathsf{id}_{A}$ but $\beta\circ\alpha\neq\mathsf{id}_{A}$.

Question

From what I understand, this is asking me to find functions $\alpha$ and $\beta$ such that $\alpha$ is surjective and not injective, while $\beta$ is injective and not surjective. Given that the question clearly states that the function is on $A$, I don't see how it is possible to construct a function that is one and not the other.

Does this mean I need to define two functions as $\alpha:A\to A$ and $\beta:A\to A$ or is there another way of interpreting this?

See here: http://math.stackexchange.com/questions/507279/example-of-left-and-right-inverse-functions — uniquesolution, Sep 21 '15 at 23:19
possible duplicate: $f \circ g =\operatorname{ id}$ and $g \circ f \neq \operatorname{id}$? — robjohn, Sep 21 '15 at 23:20

score 1 · Accepted Answer · answered Sep 21 '15 at 23:19

1

Consider the functions $f$ and $g$ on the set of natural numbers $\mathbb N=\{0,1,2\dots\}$ defined as follows:

$f(0)=1,f(2)=2,f(3)=2\dots f(n)=n+1$

$g(0)=0,g(1)=0,g(2)=1\dots g(n)=max(0,n-1)$.

Then $g(f(n))=n$ for all $n$. On the other hand $f(g(n))=n$ for all $n$ except $n=0$. So $g\circ f=\mathsf{id}_{\mathbb N}$ and $f\circ g\neq \mathsf{id}_{\mathbb N}$.

Of course such a construction is possible if and only if the set is infinite.

answered Sep 21 '15 at 23:19

Asinomás

105,651

I was fixated on finite sets.. this is why I was confused. – Mirrana Sep 21 '15 at 23:24
Oh I see, for this to work you need to be able to have non-bijective functions that are injective, this is impossible on finite sets. – Asinomás Sep 21 '15 at 23:26

Give an example of two functions $\alpha,\beta$ on a set $A$ such that $\alpha\circ\beta=\mathsf{id}_{A}$ but $\beta\circ\alpha\neq\mathsf{id}_{A}$.

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