Mack
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 Feb 25 comment Mathematical Canon @shoover I believe the mathematics library at my school is in the possession of this collection! I should have a look at it. Dec 22 comment Dubiety About An Inequality Proof Whoops, he specifies that $xy > 0$ if $x \in F$, $y \in F$, where $F$ is an ordered field, and $x >0$ and $y > 0$...I reckon more careful reading elucidates all answers... Nov 22 comment Four Isosceles Trapezoids Oh, wait! When I wrote the problem statement, I forgot to write that $x$ and $y$ are assumed to be positive integers. How does that change things? Could I immediately infer that $\displaystyle A = \frac{y+x}{2} \cdot \frac{y-x}{2}$ means that $A$ has to be factored into two positive integers? Nov 17 comment Four Isosceles Trapezoids I do not understand why I have to argue that $\frac{y+x}{x}$ and $\frac{y-x}{2}$ must be integers. I want them to be integers, so I compel them to be integers; and for each fraction to evaluate to an integer, the numerator must be some multiple of $2$. Is this the sort of proof you had in mind? If not, I do not see how it is possible to prove that they must be integers. Jul 15 comment Proving the Well-Ordering Property Why not? You just do the same comparison that I did in the case of two and three elements. Jul 15 comment Proving the Well-Ordering Property But I am proving that $S$ is well-ordered, and eventually, after having added enough elements to $S$, won't I get that $S = \mathbb{Z}^+$? So, because the two sets are equivalent, any property that one thing possess, the other thing must possess, right? In particular, $S$ possess the property that it has a minimal element; so, by the equality, $\mathbb{Z}^+$ has a minimal element. Jul 15 comment Proving the Well-Ordering Property Why isn't it clear what $S$ is supposed to be? Didn't I specify that it was some subset of $\mathbb{Z}^+$, and only contained the two elements $a,m \in \mathbb{Z}^+$? Isn't that clear enough? All I am doing in my proof is continually adding more elements from the set $\mathbb{Z}^+$ into $S$. How is that any less clear than what you said about $A$ in your post? Jul 13 comment Proof About Division of Integers That is what I figured, that the problem was concerned with existence, but I was not exactly certain. Jul 11 comment Inductive proof of inequality $a\le ab$ for nonnegative integers I have another question, then. How does one prove that the sum of two positive integers yields a number greater than the individual numbers. Jul 11 comment Inductive proof of inequality $a\le ab$ for nonnegative integers Yes, we are dealing with $\mathbb{N}$. So, my reasoning is correct? Jun 25 comment Considering Vectors Geometrically You seem to be saying that the geometric picture follows from the way in which vector addition is defined, and the fact that $\mathbb{R}^n$ itself has a geometric interpretation? Jun 13 comment The role of 'arbitrary' in proofs I particularly like when you said "This just means we are not assuming anything about x..." That's an interesting idea. May 23 comment Fourier's Heat Law In Integral Form Why wouldn't we use Fourier's law to find the total heat flow? I used the Heat Equation (the PDE) to find the temperature distribution $T(x,t)$; then I was going to use Fourier's law to find the total heat flow. Isn't your final equation just the Heat Equation integrated over $x$? I still don't see how that would give the heat flow. May 23 comment Fourier's Heat Law In Integral Form I'm sorry, but I do not quite see how I can determine the amount of heat that has flown from (or into) the rod during a certain time interval from the final equation.. Could you help me with that? Feb 8 comment Integrating With Respect To $x$ So, it would not be correct to use the differential of $y$, $dy = \frac{dy}{dx} dx$? Dec 28 comment Satisfiability Problem: Determining Which People To Invite I don't believe that that is so. According to the answer key, which you seem to agree with, if Kanti attends the party, then Samir MUST attend--and there is no exception. But still, the answer key claims that p only if q is the same as q implies p, yet the people who answered my question given in the link state that p only if q is the same as p implies q. Essentially my question is, how can the answer key be correct, it would appear that it is incorrect. Dec 25 comment Conditional Statements: “only if” This is a very interesting answer, something of which I was seeking. Dec 24 comment Conditional Statements: “only if” Okay, I believe I am beginning to understand now. Thank you. Dec 24 comment Conditional Statements: “only if” Why introduce the negations? Isn't that what you are doing, "arbitrarily" introducing negations? Why can't the statement be translated as is? Dec 24 comment Conditional Statements: “only if” So, the word "only" introduces a negation? Why is that so?