Chubby Chef
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I don't understand how implication works in logic
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9 votes

Note that logical implication (aka ‘material conditional’) does not translate directly into our intuitive understanding of implication. We often think of implication as synonymous with causality, ...

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Graduate Commutative algebra book or lecture notes in German
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5 votes

Kunz's Einführung in die kommutative Algebra und algebraische Geometrie is fairly standard, and (as far as I can tell) corresponds quite neatly to the book by Eisenbud. If you're looking for something ...

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how to find the domain of this function:$f(x)=\frac{\ln(4-x^2)}{\ln(x+1)}$
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4 votes

At $x=0$ we have $\ln (1-0) = \ln (1) = 0$ in the denominator, which is undefined.

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In graph theory, is there a term for the value of the difference between indegree and outdegree?
4 votes

I believe this is called the imbalance of a vertex, see for example this paper: https://core.ac.uk/download/pdf/42932978.pdf Note that this is defined as indegree - outdegree, but clearly this can be ...

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How do I get $-2$ as a solution to $\frac{3x+1}{x+2}<2$?
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4 votes

In this case you can't multiply both sides by $(x+2)$ as it might be negative for certain values of $x$ and that would require reversing the inequality sign. Try multiplying both sides by $(x+2)^2$ ...

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Dividing Divergent and Convergent Sequences
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3 votes

Let $b_n$ converge to $l_1$, and suppose $\frac{a_n}{b_n}$ converges to $l_2$. Then by the Algebraic Limit Theorem we have$$\lim_{n\to \infty}b_n\cdot \frac{a_n}{b_n} = l_1l_2.$$ But $$b_n \cdot \frac{...

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Does limit exist at zero?
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3 votes

$$\lim_{x\to 0} \lvert \ln (x) \rvert = \infty$$ $$\lim_{x\to 0^+} \frac{1}{x} = \infty$$ Whether this limit really exists is a matter of convention. Remember that $\infty$ is just a convenient ...

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Set proving with multiple assumptions
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2 votes

Suppose for contradiction that $A\cap E \neq \varnothing$. Then there exists some $x\in A\cap E$. By definition, $x\in A$ and $x\in E$. But since $B\cap E = \varnothing$ and $A\cap C = \varnothing$, ...

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Why is Strömberg Wavelet section in 2018 book an amost exact copy of 2016 Wikipedia article?
2 votes

I have never heard of this book's publisher and, searching for the authors, couldn't find a university page or anything (as a matter of fact, this and one other book seem to be their only meaningful ...

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What type of logical statement is this statement and how do you prove it?
2 votes

It could be the case that you've only studied propositional (also called zeroth-order) logic so far. If so, then we may reword your statement slightly and write something like "if $p$ is even and ...

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Generalize a sum given first elements
2 votes

I'm not sure what you're trying to do with your example, but one way of expressing $f(n)$ with a general formula is $$f(n) = \sum_{i=0}^{n} (n-i+1)\rho^i.$$

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Show |x| is not differentiable at x=0 using delta epsilon
2 votes

Note that, by definition of a derivative, for $\lvert x \rvert$ to be differentiable at $0$ the limit $$\lim_{x\to 0} \frac{\lvert x \rvert - \lvert 0 \rvert}{x - 0} = \lim_{x\to 0} \frac{\lvert x \...

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Are there names for function which doesn't have all items in it's domain as defined?
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2 votes

A discontinuous function is still defined on every point in its domain, discontinuity is defined in terms of limits. What you’re describing is called a ‘partial function’, I believe the term sees some ...

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Proving whether a limit exists at a point (piece-wise function)
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2 votes

So, I think the misunderstanding you have is that $\varepsilon - \delta$-type proofs cannot be applied to one-sided limits. The first step, when given a function as simple is this, is to sketch it. ...

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How does $1.88x < z <2x$ imply $z>6$?
2 votes

You've already essentially answered your own question, although the lower bound you give for $z$ seems somewhat arbitrary. You've already realised that $x \geq 9$, so you can easily substitute $9$ ...

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One thinking problem of uncountable set in real analysis
1 votes

I believe your concern regarding the supplied proof is valid. As a matter of fact, I think the chosen proof strategy isn't the optimal one - it is often said that a direct proof is better than a proof ...

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cardinality of power set of the following set
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1 votes

As stated, the problem doesn’t make sense because $6(3)^6 = 4374 < 6561$ and $6(3)^7 = 13122 > 6561$. Taking the logarithm you’d expect $T_n$ to be about the $7.6$th term, but that’s of course ...

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How can I prove that every word has unique representation?
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1 votes

Hint: if you're at all familiar with logic, this proof is (perhaps superficially) very similar to a standard proof of the unique readability theorem in propositional calculus. So, the first step is ...

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Determine the exact point(s) of intersection between f(x)=x^2-x-13 and it’s reciprocal function
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1 votes

The reciprocal of a function $f(x)$ is simply $\frac{1}{f(x)}$. So let's denote it by $r(x)$ for convenience. Then, to find the point(s) of intersection, you want to find all $x$ that would satisfy $f(...

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Prove that there exists N ∈ (N) (naturals) such that $a_n$ > 0 for all n ≥ N.
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1 votes

I don't think your proof is quite right. Instead, try expanding the definition of a limit a little bit. The key will be to pick some $\varepsilon >0$ that is strictly less than the limit $L$, say $\...

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What does it mean to equate the coefficients of like terms when solving for A and B in partial fractions?
1 votes

I think you're a little confused about the steps in this problem. Note that, after you've multiplied both sides by the denominator, you need to try and solve the resulting equation, in this case $$7 = ...

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The range of an inverse function
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1 votes

Two ways to look at it: one is that given a (bijective) function $f\colon X\to Y$ we define the inverse to be a function $f^{-1}\colon Y \to X$. But your $f$ is a function of the form $1/k$ and this ...

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Books like Stephen Abbott's Understanding Analysis
1 votes

This is a late answer ($3$ years late!), but I thought I'd put my two cents in just in case somebody ever stumbles across this question. One book that matches your description perfectly (and, I'd ...

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Are my examples of False $\implies$ False being true correct?
1 votes

Your first example is technically correct. I say technically because there seems to be a subtle difference between the actual definition of $\implies$ and the way most mathematicians use it. Of course,...

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I found $\lim _{x\to \:0}\left(1-\frac{1}{x}\left(\sqrt{1+x^2}-1\right)\right)$ to be -infinity
1 votes

Using the standard limit properties: $$\begin{align} \lim_{x\to 0} (1 - \frac{1}{x}(\sqrt{x^2+1}-1)) &= \lim_{x\to 0}1 - \lim_{x\to 0} \frac{\sqrt{x^2+1} - 1}{x} \\ &= 1 - \lim_{x\to 0}\left(...

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Is Big-O closed under composition?
1 votes

You’ve got the right idea, but the notation is a little confused. Using your counterexample, let $g(x) = f(x) = x^2$. Trivially, $f, g \in O(h(x))$ where $h(x) = x^2$. Naturally, $f \circ g (x) = x^4$,...

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How do you find a square on a grid from a position?
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1 votes

The simplest method would be to use the ceiling function: that is, $\lceil x \rceil$ is the smallest integer which is not smaller than $x$. So, supposing the squares are numbered $1$ to $10$, a pair ...

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Are there functions with a constant output regardless of input, or functions whose input is limited to a single number? (Cartesian coordinates)
1 votes

A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:\{1\} ...

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Quantifiers and variable link
1 votes

I’ll start with the second statement, as I think it makes more sense that way. $\exists x, \forall y, P(x,y)$ means that for all $y$ there exists such an $x$ that the predicate $P(x,y)$ is true. ...

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Ron and Matt traded postcards. Ron traded half . Matt traded 9. How many postcards did they have before the trade?
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1 votes

Let $x$ stand for the amount of postcards that Ron had before trading, and let $y$ stand for the amount of postcards that Matt had before trading. So, to find $x$ and $y$ we can form simple equations ...

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