Tagged Questions

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How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
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How to prove a function from a set of triples to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection?

Let $Y=\{y_1, y_2, y_3, y_4,y_5\}$ The function from the set of triples $(y_{i_1},y_{i_2},y_{i_3})$ where $i_1 \le i_2 \le i_3$ to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection given by ...
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How to handle a function from a set of functions to another set of functions?

Given sets $X$ and $Y$ we denote the set of functions from $X$ to $Y$ by $\text{Fun}(X,Y)$. Let: $k,n \in \mathbb{Z}^+$ $X_1 = \{x_1,x_2\dots, x_{k+1}\}$ $Y = \{y_1, y_2, \dots , y_n\}$ Then, ...
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Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases}$$ Let $y \in \mathbb{R}$. How would I prove that there ...
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Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
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Help to prove the existance of a function

Let $f:X \rightarrow Y$ be a function. Prove that there exists a function $g:Y \rightarrow X$ such that $f \circ g = I_Y$ if and only if $f$ is a surjection. I need help on proving the following: ...
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Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
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Help to prove $f$ is surjective $\Leftrightarrow \forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset$

Let $f:X \rightarrow Y$ be a function with graph $G_f \subseteq X \times Y$. Prove that $f$ is surjective if and only if $\forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset$ I have some ...
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Lowest possible value of a function with derivative greater than 2

I have the following two problems, and want to attempt to solve them with Mathematical rigour(which I don't yet possess): Suppose that $f$ is differentiable on $[1,4]$ and is such that $f(1) = 10$ ...
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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.
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Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
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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, $\mid B \mid^{\mid A \mid}$ ...
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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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Do you need to find the domain of a function to prove injectivity?

I teach math and have had this debate with a fellow teacher. I believe the domain of a function is really not important to determine if a function is or is not injective if there is no "real life" ...
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I tried to prove this but I am not sure if its correct. Please help me out with any tips or advice on how to improve. Here it is: First let $f(A\setminus B)\cap f(B)=\emptyset$. Now $$f(A\setminus ... 1answer 33 views h \circ g  is 1-1 and onto g is 1-1 and onto h is onto - how to prove that h is 1-1 too? h \circ g  is 1-1 and onto g is 1-1 and onto h is onto I am trying to prove that h is 1-1 too. h \circ g(x1)  =h \circ g(x2) \rightarrow x1=x2  And due to the fact that g is ... 1answer 47 views Surjection Equivalence [duplicate] I'm trying to show that if a function f:X\to Y is surjective it's equivalent to saying that f \left(f^{-1}(B)\right) = B for each B \subseteq Y. The definition of surjective that I'm using is ... 3answers 145 views Proof that \frac{(x+y)-abs(x-y)}{2} equivalent to \min(x,y) I plotted the two functions \frac{(x+y)-abs(x-y)}{2} and min(x,y) in the range [-1, 1] and they look the same. The both min and abs functions are defined as expected. ... 2answers 76 views Verification of Proof of a Bijection from A to B Problem: For  a,b \in \textbf{R} with  a < b, prove an explicit bijection of  A = \{ x : a < x < b \}  onto  B = \{ y : 0 < y < 1\} . My attempt: We consider  f(x) = ... 1answer 47 views Verification of proof that f(x) = \frac{x-a}{b-a} is bijective over the reals We consider  f(x) = \displaystyle \frac{x-a}{b-a}  for f: \textbf{R} \rightarrow \textbf{R}  where a,b are both constants such that a,b \in \textbf{R}  and b-a \neq 0. Proof that  f is ... 1answer 56 views Method for proving  f^{-1}(G \cup H) = f^{-1}(G)\cup f^{-1}(H)  Prove that if  f: A \rightarrow B  and  G,H  are subsets of  B , then  f^{-1}(G \cup H) = f^{-1}(G)\cup f^{-1}(H) . My (incorrect) Attempt: Suppose  x \in f^{-1}(G\cup H) . Then there ... 1answer 26 views Prove Bijection in roots of unity function Given k \in \mathbb{N}, G_k = \{z \in \mathbb{C} |z^k =1 \} . Probe that if n and m are coprime, the function f: G_n \times G_m \rightarrow G_{mn}, f(\alpha, \beta) =\alpha\beta is bijective. ... 1answer 97 views A polynomial is called a Fermat's polynomial… A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that f(x) is a Fermat polynomial such that f(0) = ... 2answers 86 views prove that f:X\rightarrow Y is surjective if and only if f(f^{-1}(C))=C I need help with proving this: f:X\rightarrow Y is surjective if and only if f(f^{-1}(C))=C C\subseteq Y Thanks. 4answers 55 views check if f(f^{-1}(D))=D I have to check whether f(f^{-1}(D))=D. I think this is not true but I'm stuck in my proof. Can somebody help me? Thanks in advance. 2answers 364 views proving whether a function is one-to-one/onto 1) f(n) = 2n + 1 from set of integers to set of integers 2) f(n) = 2[n/2] from set of integers to set of integers [] is floor Could someone demonstrate how I ... 0answers 41 views Prove that f: \mathbb{N} \rightarrow \mathbb{N}-\left \{ 1 \right \} given by f(x) = x+1 is 1-1 and onto f: \mathbb{N} \rightarrow \mathbb{N}-\{1\} given by f(x) = x+1 is 1-1 and onto. Proof: (1-1) Suppose f(x_{1}) = f(x_{2}) for x_{1}, x_{2} \in \mathbb{N}. Then x_{1} + 1 = x_{2} + ... 2answers 145 views Prove that the greatest integer function: \mathbb{R} \rightarrow \mathbb{Z} is onto but not 1-1 Statement: the greatest integer function int: \mathbb{R} \rightarrow \mathbb{Z} is onto but not 1-1 Proof: let x \in \mathbb{R}, then int(x) \leq x and is an element of \mathbb{Z}. Since ... 1answer 160 views Prove if f: A \rightarrow B, g: B \rightarrow C, and g \circ f: A\overset{1-1}{\rightarrow}C, then f: A \overset{1-1}{\rightarrow} B Statement: If f: A \rightarrow B, g: B \rightarrow C, and g o f: A\overset{1-1}{\rightarrow}C, then f: A \overset{1-1}{\rightarrow} B Here's my proof by contradiction. Proof: Assume f is not ... 1answer 57 views Prove that functions are one-to-one Given f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^{3} Proof: Assume f(m)=f(n) for some m, n \in \mathbb{R}. Then m^{3}=n^{3}, and m=n. f is one-to-one. Given f: \mathbb{R} ... 2answers 148 views How to prove that f(x,y)=x-y is one-to-one? Given f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, f(x,y) = x-y. How can I prove that the function is one to one? I know that f(x,y)=x-y is a plane, and just by visualizing it I can see ... 1answer 83 views For bijection f:A \rightarrow B, prove that f^{-1} \circ f = {\text {id}}_{A} I have to prove that for a bijection f:A \rightarrow B, f^{-1} \circ f = {\text {id}}_{A}, where {\text {id}_A} is the identity function of A, and we define f^{-1}: B \rightarrow A by ... 1answer 97 views Function Surjectivity Proof I have this question: Prove that a function f:X\rightarrow Y is surjective iff for any finite set Z and any function g:Z\rightarrow Y there exists a function h:Z\rightarrow X such that ... 2answers 367 views How do I prove that a function grows faster than another? [closed] I need to prove that one function, say n grows faster than say, \sqrt{n}? 2answers 59 views Why does this limit work? Let h(x)= (1+1/x)^x and g(x) be another function. Now suppose \lim\limits_{x \to \infty} g(x)= \infty. Then \lim\limits_{x \to \infty} h(g(x)) =\lim\limits_{x \to \infty} h(x)=e. I would ... 2answers 65 views Proof that a function is surjective to R I'm having difficulties proving that the function$$\frac{\sin(\frac1x)}{x^2}$$is surjective to \mathbb R. on the interval (0,10]. I tried to use the intermediate theorem, but that of ... 1answer 109 views Prove that the function f(x)=\frac{1}{x} is continuous at the point x=2. I am looking to prove that the function f(x)=\frac{1}{x} is continuous at the point x=2. So we nee that given any \epsilon>0,\ \exists\delta>0 so that |f(x)-f(2)|<\epsilon\\ whenever ... 3answers 79 views Prove f : A\rightarrow B, g: B\rightarrow C , and g\circ f: A \overset{1-1}{\rightarrow}C, then f:A\overset{1-1}{\rightarrow}B Could anyone please explain how to approach this problem, I'm honestly having a hard time figuring out where to start the problem. I know that I have to show that \forall x,y\in A , if f(x)=f(y), ... 1answer 71 views Can you prove this three-way linear map composition? OK, this was an example that my prof gave when talking about surjective, injective and bijective functions. I also am curious if I am approaching this the right way. (Everyone here has been a really ... 3answers 130 views How do I prove that the sine function on the domain [-1/2, 1/2] is injective? How do I prove that the sine function on the domain [-1/2, 1/2] is injective? I completely understand the concept but I'm having trouble writing a proof for this. Thanks in advance. 1answer 44 views Preimages of a function: Is the following proposition true or false? Let g: ℤ \times ℤ → ℤ \times ℤ be defined by g(m,n) = (2m, m – n). Is the following proposition true or false? Justify your conclusion. For each (s, t) ∈ ℤ \times ℤ, there exists an (m, n) ∈ ℤ ... 2answers 62 views prove f^{-1}(B)=A I am given A_1, A_2 \subseteq A and B_1,B_2 \subseteq B. and the function f: A \rightarrow B I want to prove that f^{-1}(B)=A. I just assume that here one is talking about ... 2answers 106 views How to exactly write down a proof formally (or how to bring the things I know together)? I've always had trouble with this: I know everything I need to know but I can't connect everything I know in the way it's being wanted from me. This is what I have to do: Prove for f : M → N the ... 1answer 70 views Proving \forall f\in \mathscr F\exists g\in \mathscr F  so that g(f(1)) = 2. Let \mathscr F denote the set of all functions from \{1,2,3\} to \{1,2,3\}. Prove or disprove. (a) \forall f\in \mathscr F\exists g\in \mathscr F  so that g(f(1)) = 2. (b) \forall f\in ... 8answers 8k views How do I prove that a function is well defined? How do you in general prove that a function is well-defined?$$f:X\to Y:x\mapsto f(x) I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
Let $[n]$ be the set of integers $\{1, 2, \ldots, n\}$. I want to find the number of onto functions from $[m]$ to $[3]$. The answer I found was $3!\cdot (3)^{m-3}$ My reasoning is: We have a ...
Let $\lbrack k\rbrack$ be the set of integers $\{1, 2, \ldots, k\}$. What is the number of one-to-one functions from $m$ to $n$ if $m \leq n$? My answer is: $\dfrac{n!}{(n-m)!}$ My reasoning is the ...