# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### How to write a proof of $A\setminus B = \emptyset \leftrightarrow A \subseteq B$

I think the best way to prove this is by contradiction, but I'm struggling with the concept of how to write it properly. $$A\setminus B = \emptyset \leftrightarrow A \subseteq B$$
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### Proving $\mathbb{Z × N}$ is countable. [closed]

How would I prove that $\mathbb{Z × N}$ is countable? The hint given was to follow to indicated order. Thanks!
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### Proving the treewidth of a graph is the maximum treewidth of the connected components

We have a graph $G = (V,E)$ and $C$ is the set of connected components of G. I want to prove that $tw(G) \geq$ Max $tw(C)$ where tw is the treewidth. I know the out sketch of the proof is to take ...
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### Proving a matrix is triangular

linear algebra proof I'm having trouble with: Let A be a square matrix. Prove that there exists a matrix $B$ so that $BA$ is a triangular matrix. I tried turning it into a homogeneous system of ...
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### Proving a Particular Coefficient of a Power Series Equals $0$

Suppose I have a particular function $$F(x,z) = \sum_{n=0}^\infty{A_n(x)\frac{z^n}{n!}}$$ and suppose, through the use of a particular computer algebra system, that the particular polynomial ...
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### Vector Proof Involving Triangle

I'm stuck on the following homework question: Given the triangle $PQR$, with $X$ placed on $PR$ dividing it into a ratio of $2:3$, and $Y$ the midpoint of $PQ$, prove that if $Z$ is the ...
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### Equality between limit and integral whose integrand diverges at some point.[Edited]

Let $f:[0,1]\times[0,1]\to\mathbb{R}\cup\{\pm\infty\}$ be a function such that, for a given point $\hat{x}\in(0,1)$, $f$ is continuous in $[0,\hat{x})\times[0,\hat{x})$ and $f(\hat{x},\hat{x})=\infty$....
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### Proving a lower bound and upper bound?

I understand why the empty set is a lower bound and A is an upper bound. The only problem I am having is putting my thoughts into a mathematical solution. Can anyone help out? Thanks. Let A be a set ...
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### Is the following proof correct?

Is the following proof correct? Let’s say we find integers $x$ and $y$ such that $x^2 ≡ y^2($mod $n)$ and $n$ has at least $2$ distinct factors not equal to $0$ or $n$. I intend to show that there is ...
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### Rational Power Question

Show that if $a ∈ Q$ is positive and if $0 < x < y$ then $x^a < y^a$. I was told to use the difference theorem for this question, but the difference theorem is only for natural numbers.
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### Prof of Reflexive, symmetric, or transitive relations

Consider the relation R on Z as: ∀m,n ∈Z, mRn ⇔ m − n is odd . Is R reflexive, symmetric, or transitive? What would the proof or counter proof be? Since R is a reflexive since m-n is linear,...
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### Proof concerning Fibonacci and recursively defined sequences

The series $(a_n)_{n\in\mathbb{N}}$ is given through $$a_1=1,\quad a_2=\frac{1}{2},\quad a_{n+2}=a_na_{n+1} \quad\text{ for } n\geq1.$$ I want to show that $$a_n = 2^{-f_{n-1}}$$ whereas $\,f_n$ ...
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### Proof of prime numbers in the form..

There exists a unique prime in the form of p^2 -1, p is just some integer with the restriction of p being greater than or equal to 2. Prove this. I understand that I am first suppose show a prime p ...
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### Prove $3+ 5 \sqrt {2}$ is irrational

Prove $3+ 5 \sqrt{2}$ is irrational. I have some ideas about this proof, but I am not quite finished. I understand being irrational means the number would not be in the form of $\frac pq$. I have ...
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Trying to make sense out of the idea that $100\%$ continuous decay is $\frac{1}{e}$, I thought about this: You can express $1+\frac{1}{x}$ as $\frac{x+1}{x}$, such that $\big(1+\frac{1}{x}\big)^x = \... 1answer 30 views ### proof the derivate of gamma function using the limit definition using$\Gamma(z+1)=z\Gamma(z)$and$\Gamma(z)=\lim\limits_{n\to+\infty}\frac{n!n^z}{z(z+1)\cdots(z+n)}$proof that $$\psi(z+1)=-\lim_{n\to\infty}\left(\sum_{m=1}^{n}\frac{1}{m}-\ln n\right)+\sum_{l=... 1answer 54 views ### The sum which gives 3^n So I have the following which I must prove :$$\sum_{(n_1,n_2,n_3)\,:\,n_1+n_2+n_3=n} \binom{n}{n_1, n_2, n_3} = 3^n$$I'm not sure where I must begin. This is a multinomial. 1answer 30 views ### Prove a property of a function H based on the definition provided Define$$H(n) = \begin{cases}{} 0 & n\leq 0\\ 1 & n = 1 \textrm{ or } n = 2\\ H(n-1) + H(n-2) - H(n-3) & n>2\\ \end{cases}$$Prove \forall n\geq 1 that H(2n) = H(2n-1) = n. ... 2answers 40 views ### Proof (a | b and a not divide b) -> a not divide (b+c) Prove \forall a\in \mathbb Z, \forall b\in \mathbb Z, \forall c\in \mathbb Z, (a | b \land a\nmid c) \rightarrow a\nmid(b + c). Maybe a gentle nudge in the right direction 2answers 44 views ### Multiplying matrices / corresponding systems of equations I'm having some trouble with a problem in linear algebra: Let A be a matrix with dimensions m \times n and B also a matrix but with dimensions n \times m which is not a null matrix. (That's ... 1answer 34 views ### Determining the exact one from all possible Jordan Canonical Forms of a matrix Here is the example I encountered : A matrix \ M\ (5\times 5) is given and its minimal polynomial is determined to be (x-2)^3. So considering the two possible sets of elementary ... 1answer 22 views ### Proof of an algebraic statement [duplicate] Let V be a n-dimensional vector space. Let's also say that we have two linear operators: A,B\in L(V) and AB=0. Then how do I prove that the sum of the ranks of operators is smaller than n, i.... 2answers 46 views ### Two matrix proofs linear algebra problem I'm having some trouble wrapping my head around: Given two square matrices A,B with dimensions n\times n and that A=I-AB : I've already proved with relative ease that A... 0answers 40 views ### If \Sigma is the splitting field for f over K and K\subseteq L \subseteq \Sigma, show that \Sigma is the splitting field for f over L. If \Sigma is the splitting field for f over K and K\subseteq L \subseteq \Sigma, show that \Sigma is the splitting field for f over L. I believe the general idea of this proof is as ... 2answers 47 views ### How to prove a statement with two “ if and only if” If H and K are subgroups of G, show that HK is a subgroup if and only if HK \subseteq KH, if and only if KH \subseteq HK. This statement confuses me. Does mean I need to prove that HK ... 1answer 53 views ### If f is one-to-one and continuous on the closed interval [a,b] then prove that f is strictly monotone on [a,b] If f is one-to-one and continuous on the closed interval [a,b] then prove that f is strictly monotone on [a,b]. So my plan was to prove this by contradiction. I'm wondering if there is a ... 1answer 32 views ### if f is integrable on [a,b] , show that \lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f If f is integrable on [a,b] , show that \lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f I proved that if f is integrable on [s,b] then f is integrable on [a,b] But how to prove the ... 0answers 60 views ### Is there any mistake in my proof? My little brother started fiddling around with his calculator, and noticed something curious:$$ \Large \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\sqrt{a \cdot \ldots}}}} = a$$So I went ahead and wrote a ... 5answers 151 views ### Prove number of handshakes between$n$people is$\tfrac{n(n−1)}{2}$by induction [closed] How do we calculate the number of handshakes between$n$people? And where do I apply the inductive step? 2answers 114 views ### Number of Taxicab routes in a triangular city I am assuming a triangle that is "almost" half a rectangular city with taxicab geometry. I am trying to find the number of paths in this triangular city. Assuming that the ride starts from the corner ... 1answer 59 views ### Suppose my progress is in Baby Rudin's chapter 4. Is it possible to discuss the uniform continuity of$x^t$without using facts in later chapters? Let$f: [0, \infty) \rightarrow \mathbb{R}$defined by$f(x)=x^t$. Prove that If$t \in (0,1]$then$f(x)=x^t$is uniformly continuous on$[0, \infty)$. If$t \in (1, \infty)$then$f(x)=x^...
Prove that for finite set $X$, the function $f:X \to X$ is surjective if and only if it is injective I have the idea of proof in my mind but find it difficult to translate it into mathematical ...
### prove that $n(n+1)$ is even using induction
the base case of $n=1$ gives us $2$ which is even. assuming $n=k$ is true, $n=(k+1)$ gives us $k^2 +2k +k +2$ while $k(k+1) + (k+1)$ gives us $k^2+2k+1$ whats is the next step to prove this by ...