# Tag Info

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The terminology varies, but this description may help. You would say that the new value is "20% less than" the old value, or that it is "80% of" the old value. The first describes the amount of change compared to the original value (the "relative change"), and the second describes the new value compared to the old value (the "change factor"). Relative ...

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Hyperbolic geometry is not really geometry on a hyperboloid. It's geometry on an infinite surface of constant negative Gaussian curvature, something which cannot be represented even in 3D. You can model it using a sheet of a hyperboloid, but the metric you get isn't the normal 3D metric you'd intuitively expect. Elliptic geometry is not the geometry on an ...

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Given a ground set $X$ and an $n\in{\mathbb N}_{\geq1}$ a function ${\bf a}:\>[n]\to X$ is called an $n$-tuple and is presented in the form $(a_1,a_2,\ldots, a_n)$. A tuple is then an $n$-tuple for some $n\geq1$ – it's the same thing as what you call a list in your question. Now I have never seen a "tuple of tuples", but would be willing to consider a ...

3

Sometimes, putting parentheses around logical statements can help make their meaning clearer if you are ever in doubt. In this instance: if (any infinite sequence in X has an adherent point in X) then (X is compact) What the first bracketed statement essentially means is "choose any infinite sequence in X and it will have an adherent point in X". Now, at ...

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This is not a mathematical term but a figure of speech. One says that A is an honest B to assert that A satisfies the definition of B. Usually the reason to emphasize honest is that A was informally called "B" earlier in the text. This is associated with a somewhat conversational style of writing. Specifically for distance functions, this word would ...

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One cannot really keep the answer to this separate from the question of what the words "set", "class", "family", etc mean -- because the question rests at least partially on a misconception. It looks like you think these are different things such that it makes sense to ask whether the thing you're looking at is one or the other. But that is often not the ...

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Terminology In ordinary mathematics (outside formal set theory), it is only convention that determines whether something is called a "family, set, collection, or class". One reason for the varied terminology is that we often begin with a collection of basic objects (e.g., real numbers). Then we form "sets" of these basic objects (e.g. sets of real ...

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A $20\%$ decrease is $80\%$ of a value. Likewise, a $20\%$ increase is $120\%$ of a value.

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A monoid is supposed to have an identity element, which is with free generation considered as the empty string. Hence your example translates to the free monoid on the letter $x$.

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This is a comment rather than an answer. Suppose you have a quadratic equation, say $$x^2-x-6=0.$$ To solve this we first assume that $x$ is a solution. From this assumption we derive that $x=-2$ or $x=-3$. So $x$ a solution $\Rightarrow x=-2$ or $x=3$. Therefore the set of numbers that solve the equation are $\{-2,3\}$ --- and we can say that $-2$ and ...

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In set theory, a set $x$ is transitive if for all $y$, if $y\in x$ then $y\subseteq x$. The (von Neumann) ordinals are defined to be the transitive sets that are well-ordered by $\in$. Thus the "$<$" relation between ordinals is simply $\in$. If $\alpha, \beta$ are ordinals with $\alpha\in \beta$, then $\alpha \subseteq \beta$. So any ordinal is a fine ...

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Not really. Matrices are maps between spaces, where columns and rows are base vectors for their spaces. On them, a scalar product has to be definable - and this just works if one can exist in a space that the other "lives in". They therefore cannot be smaller or larger, therefore have to be the same length. So no, not as a matrix. But you can define ...

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