0
votes
4answers
42 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
0
votes
2answers
179 views

Can someone please help with my inverse function and sets discrete math problem?

To save me some time writing everything out in latex, I'm adding a picture of the question and Ill try to explain what I understand for the problem. Just a heads up, I'm really not sure how to do this ...
3
votes
0answers
385 views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
1
vote
1answer
54 views

Composition of function with it's inverse on subdomains

I have a short question. We have to check the following statements and tell for which one the equal sign holds. Let $M \subset \mbox{domain } f$ and $N \subset \mbox{Im } f$. ...
2
votes
2answers
55 views

One-to-one functions between vectors of integers and integers, with easily computable inverses

I'm trying to find functions that fit certain criteria. I'm not sure if such functions even exist. The function I'm trying to find would take vectors of arbitrary integers for the input and would ...
4
votes
7answers
139 views

How should I understand $f^{-1}(E):=\{x\in A:f(x)\in E\}$?

I understand the concept, but I still can't figure out how to read the notation: $$f^{-1}(E):=\{x\in A:f(x)\in E\}$$ I understood the concept due to the examples, not with the notation. Can someone ...
2
votes
1answer
137 views

Left inverse of a function

Let $f$ be the function $f\colon \mathbb{N}\rightarrow\mathbb{N}$, defined by rule $f(n)=n^2$. Needed to find two left inverse functions for $f$. I know only one: it's $g(n)=\sqrt{n}$. Does anyone ...
1
vote
1answer
174 views

Inverse image of disjoint is disjoint?

If I have two sets that are disjoint i.e. $A\cap B=\emptyset$, and $\varphi \in C^1(U,\mathbb{R}^N)$, then are the inverse images (i.e. $\varphi^{-1}(A), \varphi^{-1}(B)$) also disjoint? My logic ...
2
votes
3answers
313 views

Inverse function requirements

Let f be an injective function, that is: $f : X \rightarrow Y$ $f(a) = f(b) \implies a = b$ Now, my question is, does the following need to hold in order for function to be injective: $(\forall x ...
1
vote
2answers
991 views

Proving a Composition of Functions is Bijective

If I have $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ is onto and $f\circ f\circ f = f$, how can I prove that $f$ is bijective? I know that I only have to prove that it is 1-to-1 because I'm ...