Tagged Questions
0
votes
1answer
38 views
Number of functions
Let $F$ denote the set of all functions from $\{1,2,3\}$ to $\{1,2,3,4,5\}$
a) Find and simplify the number of functions $f$ in $F$ so that $f(1)=4$.
b) Find and simplify the number of one-to-one ...
0
votes
1answer
64 views
I don't know where to begin with this functions question (one-to-one, onto)
a) Suppose that $f:\Bbb Z\to \Bbb Z$ is a one-to-one function. Define a function $g:\Bbb Z\to \Bbb Z$ by: for all integers $x$, $g(x)= -f(x)$. Prove that $g$ is also one-to-one.
b) Suppose $f:\Bbb ...
1
vote
1answer
45 views
Proof involving functions.
Consider two functions $f\colon A \to B$ and $g\colon B \to C$. How can I prove the following?
If $f$ and $g$ are one-to-one, then the composition function $g \circ
f$ is one-to-one.
If $f$ and ...
8
votes
1answer
163 views
Function mapping challange
For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant
mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$
and $f(k) = f(f(k + 1))$.
This is ...
2
votes
4answers
65 views
Solve for $x$: $4x = 6~(\mod 5)$
Solve for $x$: $4x = 6(mod~5)$
Here is my solution:
From the definition of modulus, we can write the above as $ \large\frac{4x-6}{5} = \small k$, where $k$ is the remainder resulting from ...
2
votes
1answer
47 views
Showing a bijection with a contraction
I have the function $F(x) = x + f(x)$ where $f(x)$ is a contraction: $|f(x)-f(y)| \leq \alpha|x-y|$ for some $0 < \alpha < 1$ and all $x, y \in \mathbb{R}$
I want to show that $F$ is a ...
0
votes
0answers
25 views
Help with functions, confirming if I'm correct.
Let $\mathscr F$ denote the set of all functions from {1, 2, 3, 4} to {1, 2, 3, ... , 10}.
a) Find and simplify the number of functions $f \in \mathscr F$ so that f(1)=1 and f(2)=2.
b) Find and ...
0
votes
1answer
42 views
Help proving and counting functions.
Let $\mathscr F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$.
a) Of the two following statements, one is true and one is false. Prove the true statement. Write out the ...
0
votes
2answers
48 views
one-, to-one, and onto functions
What determines a function as one-to-one, and onto?
And what would this function be classified as?
$A = B = \Bbb Z, f:A\to B$
$f(a) = a-1$
Little help please?
1
vote
2answers
26 views
Let (a,b) and (c,d) be intervals on R, and find an injective and surjective function from (a,b) to (c,d)
so here is this question I got stuck on:
Let $(a,b)$, $(c,d)$ be intervals (not sure if that's the correct term) on $\Bbb R$, so that $a<b$, $c<d$. Find an injective and surjective function ...
0
votes
4answers
38 views
Summation of n-squared, cubed, etc. [duplicate]
How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
-1
votes
2answers
52 views
Pigeonhole Principle
Explain the following using Pigeonhole Principle is it is true:
1) If we choose 10 points in a $3 x 3$ inch square, there must be two points of the 10 which are at distance less than or equal to ...
2
votes
3answers
72 views
Counting 1:1 and onto functions
I'm faced with the following questions:
1) How many functions are there from a set of size 3 to a set of size 5? How many of them are 1-to-1?
2) How many functions are there from a set of ...
0
votes
2answers
42 views
Determination of Functions, 1:1, and Inverse
For the following relations, I need to answer: 1) Is it a function? If not, explain why and stop. Otherwise, 2) What are its domain and image, 3) Is the function 1:1. If not, explain why and stop. ...
1
vote
3answers
54 views
Determine if function is well defined
I am having difficulties determining if the following function is well defined:
On certain computers the integer data type goes from $-2, 147, 483, 648$ through $2, 147, 483, 647$. Let S be the set ...
0
votes
0answers
13 views
Functions defined on General sets [duplicate]
I am learning how to determined whether a function is well defined. I am doing so by relying on two disticnt reasons that show a not well defined function: (1) There is no y that satisfies the given ...
0
votes
1answer
110 views
Discrete Math functions and sets
Let $A$ and $B$ be subsets of $\Bbb{Z}$, and let $F = \{f : A\to B\}$. Define a relation $R$ on $F$ by: for any $f,g\in F$, $fRg$ if and only if $f - g$ is a
constant function; that is, there is a ...
0
votes
2answers
90 views
Function and equivalence relations question
Let A and B be subsets of the set Z of all integers, and let F denote the set of all functions
f : A to B. Define a relation R on F by: for any f,g element of F, fRg if and only if f - g is a
...
0
votes
3answers
40 views
Function mapping
If there is a set $|A| = n$ and set $|B| = m$ how many functions are mapping $A$ to $B$?
It has been established that this is $m^n$.
How many of these are one-to-one?
I think this means that each ...
1
vote
2answers
58 views
Two-to-one functions
Let f be a function. We say that f is two-to-one provided for each $b \in\operatorname{im} f$ there are exactly two elements $a_1, a_2 \in\operatorname{dom} f$ s.t. $f(a_1) = f(a_2) = b$.
For a ...
-1
votes
3answers
56 views
How to find the inverse of a function $f:\mathbb{Z}_{30}\to\mathbb{Z}_{30}$ defined by $f([a])=[7a]$
If $f:\mathbb{Z}_{30}\to\mathbb{Z}_{30}$ is a function defined by $f([a])=[7a]$, show that $f$ is one-to-one and onto, and find $f^{-1}$.
I've got proof that the function is well defined, one to one, ...
1
vote
1answer
73 views
Iteration of the function $d(n)=a-n$
I start by defining the function $f$
$$f(0)=0,~~~~~f(n+1)=d(f(n))=a-f(n)$$
So:
$$f(n)=d^{\circ n}(0)$$
$f(1)=a-0=a$
$f(2)=a-(a-0)=0$
$f(3)=a-(a-(a-0))=a$
How can I find the solution of $f(x), ...
0
votes
2answers
99 views
Prove that $ f(0) \neq 0 $.
Let $ f: \mathbb{R} \to \mathbb{R} $ be a non-constant function such that $ f(a + b) = f(a) \times f(b) $ for all real numbers $ a $ and $ b $.
a) Prove that $ f(0) \neq 0 $. (Hint: Otherwise, $ f(x) ...
2
votes
3answers
104 views
Is $f:\mathbb{Z}_{30}\longrightarrow\mathbb{Z}_{30}$ defined by $f([a])=[7a]$ well defined?
To tell the truth, I'm not even sure what this means.
The professor gave an example saying that $\mathbb{Z}_m=\{[0],[1],[2],\dots,[m-1]\}$, and I sort of understand that.. but I have no idea what ...
0
votes
4answers
41 views
Find the range for $f:\{x\in\mathbb{R}\colon x\neq\frac{3}{2}\}\longrightarrow\mathbb{R}$ defined by $f(x)=\frac{4x+1}{2x-3}$.
$f:\{x\in\mathbb{R}\colon x\neq\frac{3}{2}\}\longrightarrow\mathbb{R}$ defined by $\displaystyle f(x)=\frac{4x+1}{2x-3}$.
This is for an introductory discrete math class. To tell the truth, I'm not ...
0
votes
1answer
54 views
Determining If A Relation Is A Function
I am given the simple relation $f(x)=\sqrt{x}$, where $f$ maps $R \rightarrow R$, and I am suppose to determine whether or not it is a function.
I figured that it was a function, because in the ...
1
vote
1answer
85 views
Define a $1$-$1$ onto function with domain $A$ onto the set $\{1, 2, … n\}$
Let $A = \{x^2 : x \in \mathbb{N} \text{ and } 0 \leq x^2 \leq 90\}$.
Define a 1-1 onto function with domain $A$ onto a set of the form $\{1, 2, \ldots, n\}$ to show the cardinality of $A$ is $n$.
...
0
votes
0answers
37 views
How to convert continous function into discrete function [closed]
The following functions are from a book
What is the method to convert continuous function to discrete function
...
0
votes
1answer
59 views
A discrete function and its rate of oscillation
Consider a function
$y[n]= \cos[w n ]$, where $n$ is an integer.
I have to prove that this signal will have highest rate of oscillation at $w = \pi$.
I was thinking I can take the derivatives ...
0
votes
3answers
45 views
Domain of a function is all the elements of the first set?
I am reading about functions in the textbook "Discrete Mathematical Structures" by Kolman et.al. They have given in an example that
\begin{equation}
A=\{1,2,3\} \quad\text{and}\quad B=\{x,y,z\}
...
1
vote
1answer
102 views
approximating a discrete function with a continuous one
Let $f:[0,1]\rightarrow \mathbb{R}$ be a continuously differentiable function that reaches a global maximum at $x^*\in(0,1)$. Now, consider its 'discrete' counterpart. That is, consider the collection ...
2
votes
1answer
90 views
Are there straightforward methods to tell which function has fastest asymptotic growth without a calculator?
For example, suppose I wanted to determine which of the following has the fastest asymptotic growth:
$n^2\log(n)+(\log(n))^2$
$n^2+\log(2^n)+1$
$(n+1)^3+(n-1)^3$
$(n+\log(n))^22^{100}$
Are there ...
3
votes
3answers
341 views
Domain, codomain, and range
This question isn't typically associated with the level of math that I'm about to talk about, but I'm asking it because I'm also doing a separate math class where these terms are relevant. I just want ...
3
votes
2answers
101 views
Prove $f(S \cup T) = f(S) \cup f(T)$
$f(S \cup T) = f(S) \cup f(T)$
f(S) encompasses all x that is in S
f(T) encompasses all x that is in T
thus the domain being the same, both the LHS and RHS map to the same ys, since the function ...
0
votes
3answers
102 views
Is this a valid proof of $f(S \cap T) \subseteq f(S) \cap f(T)$?
$f(S \cap T) \subseteq f(S) \cap f(T)$
Suppose there is a $x$ that is in $S$, but not in $T$, then there is a value $y$ such that $f(x) = x$, that is in $f(S)$, but not in $f(S \cap T)$. Suppose ...
1
vote
7answers
335 views
Is there any function where $f \circ f = f$ but $f(0) = 1$
Other than the identity function, is there any function where $f \circ f = f$?
$f(0)$ also has to return 1.
It must has something to do with the exponent 0 to a some coefficient...
Anyone could give ...
2
votes
1answer
42 views
Composition of two functions in $\mathbb{Z^2}\to \mathbb{Z^2}$
I need to find the composition of a function and its inverse so I have the identity function in return. My problem is that I don't seem to undestand how to proceed algebraically.
I have a function ...
0
votes
3answers
204 views
Injective & Surjective
I'm trying to do my Maths assignment, I looked at the lecturer's notes for examples but it seems like at lot of steps at skipped. Are is the example:
I understand what Injective and Surjective ...
0
votes
0answers
141 views
How do we find the inverse of a function $f(m,n)$ if there is a constant k?
I know I need to use the inverse matrice, but the problem is (the parameter) $k$, because it's a variable that can take any value depending on $k$, but it's not a variable. Think of the derivative ...
1
vote
1answer
98 views
Proof that a piecewise function is invertible.
Prove that the following function is invertible: $$g:\mathbb{Z}\rightarrow\mathbb{N}$$
$$
g(x) = \left\{
\begin{array}{lr}
-2x & : x\le0\\
2x-1 & : x>0
\end{array}
...
1
vote
1answer
82 views
Is this function $f$ onto for all positive integers?
Is this function $f$ onto for all positive integers?
$f(x) = x+2$
$\Bbb Z^+ \to\Bbb Z^+$
what about $1$?
1
vote
3answers
59 views
Binary function problem
Define $f:\mathbb{Z}\to\mathbb{N}$ by $f(n)=3n^2 + 2n + 5$. Prove that $f:\mathbb{Z}\to\mathbb{N}$ and that $f$ is one-to-one.
I have the definitions of a function and one-to-one, but I'm just new ...
0
votes
2answers
37 views
Relations problem
Can't seem to figure this one out. Could anyone help me out and explain it to me?
Thank you.
Let $P$ and $Q$ be relations on $Z$ by x$P$y iff x + 1 <= y and a$Q$b iff a + 2 <= b. Prove that P ...
1
vote
1answer
145 views
Functions and relations
I just started working with functions in my discrete mathematics class and we got presented with these two problems to think about at home. If anybody could help me out with them and explain, I'd ...
3
votes
1answer
217 views
What does the notation $|f(A)| = X$ mean?
$A$ is a set and so is $B$.
$f$ is a function $A \to B$.
I have a math question that asks about $|f(A)|$. What does the notation $|\cdot |$ mean?
5
votes
1answer
103 views
What function $f$ such that $a_1 \oplus\, \cdots\,\oplus a_n = 0$ implies $f(a_1) \oplus\, \cdots\,\oplus f(a_n) \neq 0$
For a certain algorithm, I need a function $f$ on integers such that
$a_1 \oplus a_2 \oplus \, \cdots\,\oplus a_n = 0 \implies f(a_1) \oplus f(a_2) \oplus \, \cdots\,\oplus f(a_n) \neq 0$
(where the ...
0
votes
2answers
71 views
Determining whether a function is a bijection
I am asked to determine whether a function $f(x) = x^{5} + 1$ is a bijection in $\mathcal{R}$. Proving that the function is one-to-one, I come up with the following:
$\begin{array}{lcl}f(x)& = ...
0
votes
2answers
125 views
Functions involving infinite set -> infinite set and one-to-one and/or onto
Two true or false questions:
$\mathbb{Q}^+$ means the positive rational numbers (no 0)
$\mathbb{N}$ means all natural numbers
Every function $f\colon \mathbb{Q}^+ \to \mathbb{N}$ is not one-to-one
...
1
vote
3answers
147 views
Submodularity Proof
For a fixed set $T$ and for sets $A_i ,\forall i \in \left \{ 1,2,\dots,n \right \}$ , I define $f(A_i)=\frac{|A_i|+|T|}{|A_i\cup T|}$, where $|A_i|$ is the cardinality of set $A_i$.
Is $f(A_i)$ ...
1
vote
1answer
1k views
Function which is onto and not 1-1
Here is an example of a function which I believe to be onto and not 1-1:
$f: \mathbb{N} \to \mathbb{N}$
$f(n) = \begin{cases}
0 &\text{if} \quad n=0\\
0 &\text{if} \quad n=1\\
n-1 ...
