# Tagged Questions

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### An example of a function whose domain is the set of positive integers and range is the set of integers?

I was browsing through one of my old pre-calc books, and I feel a bit ashamed to say I can't think of a simple answer. It intuitively feels impossible, as there are half as many points in the domain ...
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### How many different functions $f: A \rightarrow B$ are there if $|A| = m$ and $|B| = n$

I'm not quite sure of what this question is asking. Can someone explain please
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### show that R is an equivalent relation

Let $m>1$ be an integer, the relation $R$ on $\mathbb{N}$ given by $R=\{(a,b):a\equiv b \mod m\}$ , that is $aRb \Rightarrow a\equiv b\mod m$ where $a\equiv b\mod m$ iff $m$ divides $a-b$. Show ...
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### Prove that between every two rational numbers a/b and c/d that there is a rational number and there are an infinite number of rational numbers [duplicate]

So the full problem is Prove that between every two rational numbers $a/b$ and $c/d$ that: There is a rational number There are an infinite number of rational numbers I am having ...
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### Needed a math function, Don't know what to call it?

I need a math function $f(\ell)\to n$ whose input is a list of numbers and whose output is a noisy value (random value added to original input to get noisy output). The function $f(\cdot)$ should have ...
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### Is 'every exponential grows faster than every polynomial?' always true?

My algorithm textbook has a theorem that says 'For every $r > 1$ and every $d > 0$, we have $n^d = O(r^n)$.' However, it does not provide proof. Of course I know exponential grows faster ...
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### Comparing algorithm running times expressed in complex form

I know how to compare running times of different algorithms. Sometimes it is obvious, sometimes it requires simplifications, and sometimes dividing and using L'Hopital's rule to see if it converges ...
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### If a composition of functions is injective, must its components be injective?

Assume that we have one on one function, that look like that: $$f(g(x))$$ We need to proof or dis-proof that: I.f is one on one. II.g is one on one. I know the answer on booth questions, but my ...
### Prove Injectivity and Surjectivity of functions like $g \circ f$
I am trying to prove the following by the given: $g \circ f$ Surjective $f:A\rightarrow B$ $g:B\rightarrow C$ 1) Assuming that $g$ is Injective I want to prove the $f$ is Surjective. 2) there is ...