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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.
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accepted The role of 'arbitrary' in proofs
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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
15
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