# Tagged Questions

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### Reversible smoothing of a two dimensional function (or an image)

Smoothing of an image, or a two dimensional function is quite easy, there are many methods to achieve it, using average of near elements. But how to make it reversible? Maybe DCT (discrete cosine ...
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### How to derive this formula about the bracket function?

Is there a direct way of proving that $$[nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
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### Max Function Notation [duplicate]

I've been asked whether the following is always, never or sometimes true: $f(n) + g(n) = \theta(\max(f(n), g(n)))$ I understand the definition of theta notation, but I'm not sure how to read the ...
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### 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 ...
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### Prove a function is surjective or injective

so I'm having trouble figuring out why this question is surjective / where $0$ comes from. $f : \mathbb{N} \longrightarrow \mathbb{N}$ where $f(x) = x + 1$. so given N begins from $0$ it goes: ...
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### Classifying relations as functions

so I'm having a bit of trouble actually working out the relation here, I'm fine with determining whether its a function/partialfunction etc I just don't understand how to produce the relation and get ...
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### Trying to understand how many functions there are from A to B.

I'm trying to understand why there are $B^A$ functions from $A$ to $B$. If $A=${$a,b$} and $B=${$1,2,3$} then the functions from $A$ to $B$ are $f(a)=1$, $f(b)=1$, $f(a)=2$, $f(b)=2$, $f(a)=3$, ...
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### Prove that function f is injective, if function (g o f) is injective as well

I'm sort of stuck with this type of proof. Not quite sure how to go about it. I was wondering if someone could help me out to get started with it. I guess that the two hypotheses I have are these: ...
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### Trouble understanding Big O notation for a sum of n integers [duplicate]

This problem is an example in a Discrete Math textbook. How can big-O notation be used to estimate the sum of the first n positive integers? Solution: Because each of the integers in the sum of the ...
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### Function form understanding

I dont understand what is the function when im given this kind of form: f= {<1,1>,<2,3>,<3,2>}. I understand functions when they are given in lambda form. For example how can I find the ...
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### Defining a bijective function from $2\mathbb{N}$ to $3\mathbb{Z}-1$?

$2\mathbb{N}=\{2n:n\in\mathbb{N}\}$ and $3\mathbb{Z}-1=\{3n-1:n\in\mathbb{Z}\}$ Work: So far, my plan is to first define a bijective function from $2\mathbb{N}$ to $\mathbb{N}$ and then define ...
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### Proving that function $f:[0,\infty)\rightarrow [0,\infty)$ defined by $f(x)=\frac{x^2}{1-x}$ is bijective.

I am having a bit of trouble with the algebra for proving that the function is injective. Basically I set $f(a)=f(b)$ for $a,b\in[0,\infty)$ and $a,b\neq 1$. ...
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### Bijection of a function.

Define the function f: $(2,\infty) -> (-\infty,-1)$ by $f(x)= \frac{-x}{x-2}$. Show that f is bijective. I know i need to prove both injective and surjective, and I was able to solve the equation ...
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### 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 ...
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### Proving that a function from $N\times N$ to $N$ is bijective.

I am stuck on this problem: Define $f: N\times N \rightarrow N$ by $f(i,j)=\frac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically ...
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### 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 ...
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### 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 ...
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### Proving that $f$ is a bijection from $N$x$N$ to $N$.

I am having trouble with the following problem: $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent. Work: I ...
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### Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
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### Why is the inverse of this function not a function?

Why does $F^{-1}$ need to be defined on all of $Y$? I can have this function: $g(x)=x,\quad x\ne 3$ and even though it is not defined for all $x$ in its domain, it is still a function, right?
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### How can a function not be one to one and be a function?

My understanding of the definition of a function Given any x, there is only one y that can be paired with x My understanding of a 1 to 1 function Given any y, there is only one x that can be paired ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $A$. ...
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### 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 ...
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### 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 ...
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### Confusion surrounding functions

Hey there Mathematics, Slightly confused over some of the things in my quiz and was wondering if I could get an explanation: I thought with the first question that it's just one-to-one from X to Y ...
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### 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 ...
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### Help with Functions in Discrete Mathematics

I am having trouble solving this problem: let $p$ be a positive prime number and let $f:Z_p -> Z_p$ be defined as $f([x])=[x^2]$. Show that $f$ is a function. Give examples of how it is not ...
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### Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
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### Give a Bijection that is in-between 2 intervals and use a formal proof to show that it is a bijection. [duplicate]

∀w,x,y,z ∈ R, w < x and y < z. Given that information, supply a bijection between the two intervals. (w,x) and (y,z) Then after you find the bijection, provide a formal proof that what you found ...
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### Let A = {1,2,3,4} Let F be the set of all functions from A to A. (check the parts)

Let $\operatorname{S}$ be a relation on $F$ defined by: $\forall f, g \in F, f\,\operatorname{S}\,g \iff f(i) = g(i), \exists i \in A$. (a) Recall that the identity function $I_A : A \mapsto A$ is ...
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### How many functions with A having 9 elements and B having 7 elements have only 1 element mapped to 7?

So, the question I have is: How many onto functions [9] --> [7] have only one element mapped to 7? This is asking how many functions with A having 9 elements and B having 7 elements have only 1 ...
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### Prove that this function defined by f(x) is bijective.

Prove that the function $f: \mathbb{R} - \{1\} \to \mathbb{R} - \{1\}$ defined by $f(x)$ is bijective. $$f(x)=\left({x+1\over x-1}\right)^3$$ I am taking my first Computer Science ...
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### Proving a function is a one to one correspondence

I understand that to show a function is a one to one correspondence, you have to show that the function is both one to one and onto. Proving a function is one to one seems simple enough. However for ...
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### Discrete math functions help?

I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test? ...
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### Function Mappings

There are two “shift functions” mapping $\Bbb N$ into $\Bbb N$: $f(n)=n+1$ and $g(n)=\max\{0,n-1\}$ for $n\in\Bbb N$. How to show that $(g\circ f)(n)=n$ for all $n$, but that $(f\circ g)(n)=n$ does ...
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### Prove that a function is one to one without graphing

I know that you can prove a function is one to one by graphing it and using the horizontal line test. But in my notes it showed another way to prove a function is one to one but I am not sure if I am ...
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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