# Tagged Questions

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### Prove that a function defined on points in a plane is zero

Let $n\ge3$ be an integer, and $f:P\to\mathbb R$ be a function defined on any point in the plane $P$, with the property that for any regular n-gon $<A_1A_2A_3\cdots A_n>$, ...
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### Suppose the function $f: X \rightarrow Y$ is onto. Prove or disprove that the induced map $\bar f^{-1}: P(Y) \rightarrow P(X)$ is onto

Suppose the function $f: X \rightarrow Y$ is onto. Prove or disprove that the induced map $\bar f^{-1}: P(Y) \rightarrow P(X)$ is onto. This is a powerset ---> $\bar f^{-1}: P(Y) \rightarrow P(X)$ ...
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### Some confusion in notation of Bernstein-Schroeder Theorem

Here on page 232-233 the author offers a proof of the Bernstein-Schroeder Theorem. He uses the subset $$\bigcup_{k=0}^\infty(g\circ f)^k(A-g(B))\subseteq A$$ and I'm not exactly sure how to parse ...
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### Proving that $f(x) = \cos(x)\implies f'(x) = -\sin(x)$ using the definition of a derivative

I'm having trouble grasping the concept which proves that the derivative of $f(x) = \cos(x)$ is $f'(x) = -\sin(x)$. It needs to be proven using the definition of a derivative--and I can't quite piece ...
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### $f_1,f_2 :\mathbb{R}^2\rightarrow \mathbb{R}^2$ be two functions such that…

Let $f_1$ and $f_2$ be two functions on $\mathbb{R}^2$ defined as : $$f_1(x,y)=(x+1,y+3)\\ f_2(x,y)=(x-3,y-2)$$ Which of the following are true? For any positive integer $k$ there exists a ...
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### 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 ...
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### check my short simple proof - Functions are of same magnitude. Asymptotic notation.

A simple question with a short solution I thought of, but I would like verification. $f(n)$ is a function that approaches infinity as $n$ approaches infinity. We are asked to show that ...
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### 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 ...
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### 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 ...
### 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 ...