Elementary questions about functions, notation, properties, and operations such as function composition.

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0
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2answers
41 views

Show that $A^{(x,y)}$ is countable.

Question: Let $A$ be a countable set $A^{(x,y)}$ the set of all functions from $(x,y)$ to $A$. Show that $A^{(x,y)}$ is countable. My attempt: By proposition 7.1.2iii, $\mid B \mid^{\mid A \mid}$ ...
3
votes
1answer
40 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
1
vote
1answer
34 views

Detail about in a proof

Proposition: proof: note: I don't figure it out. I'll use u' and v' instead of and . $u' \circ v'= u' ( v' ( f ) ) = u' ( f( v ) ) = f ( u ( v ) ) =f \circ u \circ v $ Something seems wrong. Can ...
1
vote
2answers
33 views

Find the value of $x$ for which $ff=gf$.

Functions $f$ and $g$ are defined by $f:x \mapsto \frac{1}{2x+1}$, $x \neq \frac{-1}{2}$ and $g:x \mapsto x+1$. Find the value of $x$ for which $ff=gf$. So I started in this way: $f[f(x)]=g[f(x)]$ ...
1
vote
1answer
16 views

Help with composite identity functions in discrete mathematics

I am having trouble with the following problem: For nonempty sets $A$ and $B$ and functions $f : A \rightarrow B$ and $g: B \rightarrow A$ suppose that $g \circ f = i_A$, the identity function on ...
0
votes
1answer
39 views

Are the two integrals equivalent?

Consider $x \in \mathbb{R}$, $A\subseteq \mathbb{R}$, $f(x)$ continuous in $\mathbb{R}$ and the integral $$ g(A):=\int_{x \in A}^{} f(x) dx $$ Is $g(A)$ equal to the integral $$ ...
0
votes
0answers
23 views

Function inverse mapping [0, +inf) to [0, 1)

I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$. It is ...
1
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3answers
43 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
2
votes
2answers
50 views

Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
0
votes
1answer
22 views

Discontinuity of the indicator function

Consider the function $q(x,\theta)=1\{ x \in \{x \text{ s.t. } \theta+x_i>0 \text{ }\forall i \}\}$ where 1 is the indicator function taking value 1 if the condition inside $\{ \}$ is satisfied and ...
1
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3answers
98 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
2
votes
2answers
63 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
0
votes
2answers
93 views

Meaning of $x^x$? [closed]

may be this question doesn't have any specific answer but I would really appreciate if people can give their views about it. I was asked: What do you mean by $x^x$? And I couldn't say anything, what ...
0
votes
3answers
21 views

Show that the mapping is one to one iff $|f(S)|=|S|$, and the two definitions for being onto are equivalent.

Definition: $f(X)=${$f(x)|x\in X$}, "$|A|$"represents the number of elements in the set A. In the title, $f:S\to T$, "$iff$" means "if and only if".$S$and $T$ are finite sets. Two definitions for ...
0
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2answers
48 views

How to prove that if $A\subseteq B$and $|A|=|B|$, then $A=B$

Apart from the question in the title, the other question that related to the first question: Define: $f(X)=${$f(x)|x\in X$}. if $X$ is finite, $f(X)\subset X$ and $f$ is one to one, then $|f(X)|=|X|$, ...
0
votes
1answer
7 views

Does each element in domain need result for onto functions?

For onto functions, do all the elements in the domain have to give a result from the range? I know that for one-to-one, every single $x$ must give a result, and one that is a unique $y$. For onto ...
0
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0answers
20 views

Discrete Math identity function proof

Hi I am having trouble with this question: For nonempty sets $A$ and $B$ and functions $f:A\rightarrow B$ and $g:B\rightarrow A$ suppose that $g\circ f=i_A$, the identity function on $A$. How do ...
1
vote
1answer
28 views

Help with identity functions in discrete mathematics

I have trouble with trying to solve the following problem: For nonempty sets $A$ and $B$ and functions $f:A\rightarrow B$ and $g:B\rightarrow A$ suppose that $g\circ f=i_A$, the identity function on ...
-1
votes
3answers
41 views

Proving functions are injective and surjective

I am having trouble with the following problem: For nonempty sets $A$ and $B$ and functions $f:A \rightarrow B$ and $g:B \rightarrow A$ suppose that $g\circ f=i_A$, the identity function of $A$. ...
0
votes
3answers
25 views

Injective and Surjective Function Examples

I am having trouble with this problem: Give an example of a function $f:Z \rightarrow N$ that is a. surjective but not injective b. injective but not surjective Work: I came up with examples such ...
1
vote
4answers
40 views

Proving that a function is bijective

I have trouble figuring out this problem: Prove that the function $f: [0,\infty)\rightarrow[0,\infty)$ defined by $f(x)=\frac{x^2}{2x+1}$ is a bijection. Work: First, I tried to show that $f$ is ...
0
votes
1answer
13 views

Proving Integer Modulo is Well-Defined

I have trouble figuring out this problem: $h: Z_4 \rightarrow Z_6$ by $h([a])=[3a]$ for each $a\in Z$. Prove that h is well-defined thus it is a function and that h is neither injective nor ...
1
vote
1answer
31 views

Find the fuction $g$.

If $f:x \mapsto x^2 + 3$, find function $g$ such that $gf:x \mapsto 2x^2 + 3$. I don't know how to do it, there is no such example in my book. Help?
1
vote
1answer
22 views

Series representation of a function. Generating the series formula.

in general say the question is to find the series representation of $ arctan(3x)$ the solution is the $\int \sum (-1)^n * (3x)^{2n}=$ $$ \sum (-1)^n * 3x^{2n+1}/(2n+1) $$ but my confusion is why ...
0
votes
1answer
21 views

Limits: $\lim_{x\to0^+}xe^\frac{-1}x$ and $\lim_{x\to0^-}xe^\frac{-1}x$

$$\lim_{x\to0^+}xe^\frac{-1}x \text{ and } \lim_{x\to0^-}xe^\frac{-1}x$$ Can anyone help me how to find these limits? Thanks
4
votes
3answers
44 views

Concept of a function and Idea of a formula as a function; History of

Enderton Elements of Set Theory, p. 43 (1977, Academic Press), writes: There was a reluctance to separate the concept of a function itself from the idea of a written formula defining the function. ...
-3
votes
1answer
28 views

Can I have an exponential function such that if x = infinity, y = 100?

I tried the most basic y = 100*constant^(1/x) assuming that 1/x = 0 when x is infinity, but it doesn't seem to work. This gives me a function that starts with a higher value of y and goes down till ...
0
votes
1answer
22 views

drawing smooth and continuous curve

I have been asked to draw a curve that is "smooth and continuous" which passes through (5,0) and the domain is [-1,5) and range is (-infinity, 6] No function is given just these conditions How ...
0
votes
1answer
37 views

Bijection and image

Let f: A -> B be a bijection, so f^-1: B -> A is a function. Let X be a subset of A. How do I prove that Im(f)(X) = Preim(f^-1)(X)? Thank you.
2
votes
3answers
52 views

Determining injectivity and surjectivity

Are these functions injective or surjective? Also, how should I go about proving this? The function maps $ℕ×ℕ$ to $ℤ$. $f(a,b) = 4a+5b$ $f(m,n) = m^2-n$ $f(p,q) = 5^p·3^q$ Thanks!
1
vote
1answer
65 views

Example of two finitely isomorphic structures (or $\omega$-isomorphic) that are not partially isomorphic?

One can define several maps between given structures in order to portray similarities or differences. The concept of isomorphism displays a deep structural overlapping, while for instance an ...
0
votes
0answers
19 views

Prove that $f[f^{-1} f[X]]] = f[X]$

I'm trying to prove that $f[f^{-1} f[X]]] = f[X]$, where $f: A\to B $ and $ X \subset A$. I have already proved that $X \subset f^{-1}[f[X]]$. My thoughts: First, I know that $ f[X] = \{f(x):x \in ...
1
vote
2answers
29 views

Inverse modulo function

How can we calculate the inverse of a modulo function, now I have a problem given me $f(n)=(18n+18)\mod29$, need find inverse of $f(n)$ ? how is the process to do it?
0
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0answers
13 views

A function to fit a certain S-shaped curve

I am looking for a function to fit a certain type of S-shaped curve. Here are my criteria: The curve always pass three points (0,0), (0.5,0.5) and (1,1). For 0 < x < 0.5, f(x) < x; for ...
0
votes
1answer
19 views

How many functions defined on $n$ points are possible if each functional value is either $0$ or $1$?

How many functions defined on $n$ points are possible if each functional value is either $0$ or $1$? This is from the text A First Course on Probability by Sheldon Ross. The solution he ...
2
votes
3answers
83 views

How to read $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$

Can somebody explain me step by step why does this $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$ represents the set of ...
0
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0answers
23 views

A function defined by a differential rule

I have read this: Define the non-decreasing function $Z(s)$ defined by the differential rule $$dZ=\min(ds, d\Phi(s))$$ and define its approximations $$Z_n(s) = \int_0^s \min(1, ...
0
votes
1answer
67 views

Surjectivity of Composite Functions

The question I'm asking might be rather simple, but I couldn't find relevant information (maybe it's too trivial?). Here's the question that baffled me. Let $f:X\rightarrow Y$ and $g:Y\rightarrow ...
0
votes
0answers
11 views

How to find Modulation/Demodulation pairs

$y(t)$ is output and $x_m(t)$ are inputs. Let's we have relation between output and input : $$y(t)=\sum \limits_{k=1}^n h_k(t)x_k(t)$$ where $n>1$ $$\int_{a}^{b} u_m(t,z)y(t) dt=\int_{a}^{b} ...
1
vote
1answer
28 views

A problem with moments of a function

How to show that there is no continuous function $f:[0,1] \to \mathbb{R}$, that satisfies $\displaystyle \int_0^1 x^nf(x)\,dx = 0$ for all $n = 0,2,3,\cdots$, but $\displaystyle \int_0^1 xf(x)\,dx ...
0
votes
0answers
19 views

questions about Darboux theorem

i saw a proof of Darboux theorem. it says that we have function f and her derivative is f ', and f matches all the condition of Darboux theorem. but than, it says that if f '(a)=f '(b) then the claim ...
0
votes
1answer
19 views

Are there functions from $P(\Bbb Z)$ to $P(\Bbb N)$ that are bijective?

I've been trying to find a function from $P(\Bbb Z)$ to $P(\Bbb N)$ that are bijective. I've found some but all of them are surjective but not injective.
1
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2answers
19 views

A basic question on non-negativity of a function

how to prove that $f(x) = 2x\sin(\frac{1}{x}) - \cos (\frac{1}{x}) + 2$ is positive when $x\in (0,1]$. I can see that by plotting.
1
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1answer
21 views

Prove that a function is surjective but not bijective

I'm trying to prove that the function $f :P(\Bbb Z) \to P(\Bbb N)$ defined by $f(X) = X\cap \Bbb N$ is surjective but not bijective. In order to do this, I need to prove that $f$ is surjective and ...
3
votes
1answer
34 views

Separate the variables of the function $\frac{x^2}{\sqrt{x^2+y^2}}$

Is there a way to express the function $\frac{x^2}{\sqrt{x^2+y^2}}$ as the product of two functions: $f(x)\cdot g(y)$, i.e. one in each variable? This is becasue I want to apply a convolution whose ...
4
votes
1answer
52 views

How many surjective functions are there from $A=${$1,2,3,4,5$} to $B=${$1,2,3$}?

I want to find how many surjective functions there are from the set $A=${$1,2,3,4,5$} to the set $B=${$1,2,3$}? I think the best option is to count all the functions ($3^5$) and then to subtract the ...
0
votes
2answers
24 views

Question on function notation. Is this correct?

$f$ is the function $f(x)=2x+5$ a) Find $f(3)$ I know that this is $2(3)+5=11$ Express the inverse function $f^{-1}$ in the form $f^{-1}(x)=$ I dont know how to do this. Then the question goes ...
-1
votes
2answers
30 views

Prove that there is bijection between sets

I need to prove that there is a bijection between these sets: $$A = [0, 1], B = (0, 1/2) ∪ (1/2, 1)$$ I tried to use Cantor–Bernstein–Schroeder theorem but I am lost. Can you help me?
0
votes
1answer
21 views

how to define a function which returns a multiset

Given graph $G(V,E)$, how can I formally define function $f\prime$ which takes a node $v \in V$ and returns a multiset?
0
votes
0answers
20 views

Limit a formula results to a values interval

I have a very simple mathematical formula that is the following: $$z=100000\cdot \frac{\frac{x}{10000} - 20}{6}\cdot(1 - (10(\frac{1}{5} - y))$$ I would like to improve the formula ensuring that the ...