0
votes
1answer
31 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
0
votes
1answer
31 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
1
vote
0answers
35 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
1
vote
0answers
70 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
1
vote
1answer
34 views

Prove that $ f\left(\bigcup _{i\in I}A_i\right)=\bigcup_{i\in I}f(A_i)$

Let $f \colon A\rightarrow B$ be a function. Furthermore let $I$ be a set and $\forall i\in I,A_i\subseteq A$ (1) $\displaystyle f\left(\bigcup_{i\in I}A_i\right)=\bigcup_{i\in I}f(A_i)$ (2) ...
0
votes
1answer
57 views

Use induction to prove that a function is not one to one

Suppose that m and n are positive integers with m > n and f is a function from $\{1, 2,\ldots, m\}$ to $\{1, 2, \ldots , n\}$. Use mathematical induction on the variable n to show that f is not ...
0
votes
1answer
31 views

Is the following claim true? (about a function that is strictly increasing and surjective)

We have a function $f$, $f:I \to F$, where $I$ is an open-ended interval, and $F$ is also an interval. $f$ is strictly increasing and surjective. I was trying to prove the fact that $f$ is continuous, ...
2
votes
1answer
77 views

How to prove a function is not onto?

Let $f : Z\to Z$ be the function defined by $f(x) = 3x + 1$. Prove that $f $ is not onto, using a proof by contradiction. (Choose an integer $n$, and then prove ($\forall m \in Z$)($f(m) ≠ n$) by ...
0
votes
5answers
66 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
2
votes
1answer
37 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
0
votes
1answer
44 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
3
votes
2answers
161 views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
4
votes
2answers
88 views

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. [duplicate]

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. EDIT: Actually, this identity should hold even if $f$ is not one-to-one (injective), right? ...
0
votes
2answers
56 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
0
votes
2answers
93 views

Prove that a function is bijective

So, the problem sounds like this. You have two bijective functions $f:\mathbb{N} \to A$, $g:\mathbb{N} \to B$. We define the function $ h:\mathbb{N} \to A \cup B $, defined as: $$ h(n) = ...
1
vote
0answers
16 views

Existence of a particular transformation

I've a set of data points $S = \{ x | x\in [0,1]\}$ (i.e. real values from the unit interval). In some cases I've big clusters in the data and I want to spread the values in between the unit interval ...
2
votes
1answer
67 views

Help needed - Approximating functions with geometric integration and derivation

I've somehow managed to approximate some functions using cheap tricks as geometrically derivating the function and then geometrically integrating an easier equivalent of the derivative (see here for ...
1
vote
2answers
30 views

Bijection and it's inverse

Given $f: X \to Y$ such that $f$ is a bijection prove the existence of a $g:Y\to X$ such that: $f \circ g = 1_Y $ and $g \circ f = 1_X $ Now since $f$ is bijective $\forall y \in Y: \exists!x ...
1
vote
4answers
110 views

If $f \colon A \to B$, $g :\colon B \to C$ and $g\circ f \colon A \to C$ are bijections. Prove that $f $ is 1-1, $g$ is onto.

From what I understand, one-to-oneness means every element in $A$ is mapped to a unique element in $B$. To be onto, means for every $y$ in $B$, there exist at least one $x$ in $A$ from which it can ...
4
votes
2answers
67 views

Assuming that A and B are non-empty, if there is an injective function F : A -> B then there must exist a surjective function g : B -> A

Either give a counter-example, or a proof. A question in my proofs review. From what I understand we must assume each element of A is carried to a unique element of B (i.e. every value of A is ...
1
vote
3answers
35 views

Help with this function proof

If a function is bijective then its inverse is unique. I came across this in my textbook and was wondering how it is proved. Thank you.
0
votes
2answers
46 views

Integrable functions and absolute values

I have qutoted that the absolute value of an integral is less than or equal to the integral of an absolute value of a function. I have also said $|-g(x)| \le g(x) \le |g(x)|$ implies the integral ...
2
votes
3answers
32 views

Show that $x_0$ must be an integer. Conclude that $\sqrt[n]{2}$ is irrational for every $n \geq 2$

I have a problem in my workbook that is as follows: Let $f = x^n + a_{n-1}x^{n-1}+\dots+a_1x+a_0 = 0 $ with $a_i \in \mathbb{Z}$. Suppose there exists a rational number $x_0$ with $f(x_0) = 0$. ...
0
votes
1answer
85 views

Beginner proof of image of functions and functions of sets

This is the third time I got my proofs handed back from my teacher. She won't tell me what's wrong except I have to redo it. I am running out of luck and I need help towards the right direction! The ...
2
votes
5answers
177 views

Proof strategy for $(\Leftarrow)$: If $g \circ f = id_A$, then $f$ onto $\Leftrightarrow$ $g$ 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets $A$ and $B$ and functions $f \colon A \to B$ and $g \colon B \to A$, suppose that $g \circ f =$ the identity function on $A$. $(♦)$ (e) $(\Leftarrow)$ Assume that $g$ is ...
1
vote
2answers
96 views

How to prove a function from $\mathbb N\times \mathbb N$ to $\mathbb N$ is bijective. [duplicate]

I am having trouble with this problem: $f\colon \mathbb N\times \mathbb N \rightarrow \mathbb N$ is defined by $f(i,j)=\dfrac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection from ...
0
votes
0answers
45 views

Proving that the set of irrational numbers is uncountable [duplicate]

Work: Assume that the set of irrational numbers is countable. Since $Q$ is infinite, it is therefore denumerable. Therefore, there exists a bijective function $f: N \rightarrow Q$. From here I am ...
1
vote
1answer
48 views

Proving that $f: N\times N \rightarrow N$ is surjective [duplicate]

I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective. Work: I know that using the theorem in which $n$ is the product of prime numbers ...
1
vote
2answers
74 views

Proof strategy for $(=>)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist ...
1
vote
1answer
42 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 ...
1
vote
1answer
56 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
84 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 $A$. ...
1
vote
4answers
46 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
20 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
2answers
27 views

Help with Discrete Math Functions and Bijections

I have trouble with the following problem: Prove that the function $f(x)=x^2-2x+3$, with domain $x\in (-\infty, 0)$, is a bijection from $(-\infty, 0)$ to its range. Work: I tried to first prove ...
0
votes
2answers
32 views

define $f :R\to R$ by $f(x)=\frac{1}{(x-1)}$ when $x<1$ and $f(x)=\sqrt{(x-1)}$ when $x\geq 1$. Show that $f$ is a bijection and determine its inverse

A bonus Q on a discrete math/proofs test, I know I must prove injectivity and surjectivity, but am not exactly sure how to do so. Please help, this will be covered on the upcoming final exam in April. ...
0
votes
1answer
52 views

Let $X \neq \emptyset$, define the relation$A\sim B$ if there exists a bijection $f : A \to B$, Show that $\sim$ is an equivalence relation on $X$.

A question on my last proofs midterm, I know I must prove injectivity and surjectivity, but there aren't really any obvious conditions or descriptions on S that helped me to manipulate it to try and ...
1
vote
0answers
150 views

For the non-empty sets A, B and C, let $f : A \to B$ and $\,g : B \to C$. Prove or disprove the following statements:

(a) If $f$ is onto then $g\circ f$ is onto. (b) If $g$ is onto then $g\circ f$ is onto. (c) If $f$ is one-to-one then $g\circ f$is one-to-one. (d) If $g$ is one-to-one then ...
0
votes
2answers
54 views

Consider the function h where $h(x,y) = (x+y,x-y)$, $h : \mathbb N\times \mathbb N\to \mathbb N\times\mathbb N$ [duplicate]

Is the function h onto and one to one? Prove this. Online bonus question on a recent proofs quiz on the topic of one-to-one and onto functions. Gave me a bit of grief (the mapping stuff). Also ...
1
vote
3answers
115 views

Let X and Y be finite non empty sets such that $|X| = |Y|$. Show that a function $f : X \to Y$ is onto if it is one to one.

Hello this is a recent question posted on my course website for bonus marks. I am not exactly an expert at proving bijection (our current topic of study) and the definitions of onto and one-to-one are ...
0
votes
4answers
74 views

Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$

Determine if $g$ is injective; surjective; bijective. Question on a recent test regarding one-to-one and onto functions. Was very difficult for me, could not even begin to answer either. This is ...
0
votes
1answer
28 views

Floor and Ceiling question

This was a homework question. I wasn't able to get far because I couldn't determine the properties of floor and ceiling functions. Any help would be awesome. $\def\lc{\left\lceil} ...
2
votes
0answers
61 views

Proof of two properties of a simple math function

I would like to define a function to evaluate the value for some entities which receive a number of "up"s ($\mathcal{u}$) and "down"s ($\mathcal{d}$). I devised the following function: ...
0
votes
2answers
78 views

How do I prove this bijection?

The number of $n$-digit binary numbers with exactly $k$ $1$s equals the number of $k$-subsets of $[n]$. I think i'm on the right track, but I'm confused on how to write how it's onto and 1-1. This ...
0
votes
2answers
83 views

Proving onto of a two variable function

So I know how to prove a function is onto if it has 1 variable. But this one has two and I'm confused about how to approach it. $f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ such that for any ...
0
votes
1answer
66 views

Proving $\left\lfloor n\frac{\log (b)}{\log (a)}\right\rfloor =\left\lfloor \frac{\log \left(b^n+1\right)}{\log (a)}\right\rfloor$

Inspired by this question, I'd like to know how one would go about proving the below more general equation? $$n \in \mathbb{N},\;a \in \mathbb{N},\;b \in \mathbb{N}$$ $$b^n+1 \notin ...
1
vote
2answers
456 views

Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $

It is known that the following holds good: $$ \arcsin x + \arcsin y \\ \begin{align} &=\arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 \le 1 \;\text{ or }\; x^2+y^2 > 1, xy< 0\\ ...
0
votes
1answer
50 views

Is This A Formal Proof $f:Z \rightarrow N ; f(z)=|z|$ is onto and not 1-1?

There will be a function $f(z)=|z|$ defined as follow $f:Z\rightarrow N$ Prove/Disprove the function is 1-1 or onto or both. Disproving 1-1 $f(-1)=f(1)=1 \rightarrow $-1 does not equal 1 therefore ...
1
vote
1answer
1k views

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 ...
1
vote
1answer
25 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...