小型SUV的末日？小型SUV价格能买到的3款紧凑型SUV

Question

A statement reads, "The points within the parallelpiped determined by $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$ are precisely the endpoints of vectors of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$ where $...$"

I am having trouble with the word precisely in this sentence and generally, probably, in any mathematical context. Does the use of the word precisely have any logical or set theoretic meaning; i.e., does the word imply a double (set) inclusion in the sentence: "points in the parallelpiped have the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$" and "points of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$ are contained in the parallelpiped"?

If so, how would I have known this? Is there an available list of such modifiers that are used in mathematical contexts?

Answer 1

(slightly edited version of a former comment)

In a comment @Dan Rust said: The meaning of the sentence doesn't change if the word is removed, but it's used for emphasis.

To add to what Dan Rust said, and to show limitations of English (and presumably other languages), the meaning of the sentence DOES change if, in addition to "precisely" being omitted, "the endpoints" is changed to "endpoints" (i.e. omit "the"). For various reasons such as this, it's helpful in verbal explanations of precise mathematical things (and here I'm using "precise" in a different way, further showing how muddy things can be) to err on the side of too much explicitness (or alternatively, provide an example that would serve to eliminate possible alternative meanings present in natural language).

Answer 2

I'm pretty sure the meaning here is that points in the parallelepiped are exactly the same as endpoints of vectors of the form...

While the word "precisely" can mean the opposite of "vaguely", there is this other common usage that is meant to signal an if-and-only-if statement, e.g., "$P$ is a point in the parallelpiped if and only if it is an endpoint of a vector of the form...". So "precisely" there could also be colloquially rendered as "no more and no less (than)".

Answer 3

The points within the parallelpiped determined by $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$ are precisely the endpoints of vectors of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c},$ where $...$

does the use of the word precisely imply that "points in the parallelpiped have the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$" and "points of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$ are contained in the parallelpiped"?

Yes, precisely! The given sentence means

• Points are within the parallelpiped precisely when they are of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$
• Points are within the parallelpiped if and only if they are of the form $\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}$
• For every point $P,$ $P$ is within the parallelpiped if and only if there exist scalars $\alpha,\beta,\gamma$ in $[0,1]$ such that $\vec{OP}=\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}.$

"Precisely when" and "if and only if" are both Mathlish for "is equivalent to".

In set notation:

• The set of points within the parallelepiped is $\;\{\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}\mid \alpha,\beta,\gamma\in [0,1]\}$
• The parallelpiped contains precisely the points in $\;\{\alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}\mid \alpha,\beta,\gamma\in [0,1]\}.$

Notice that in the bullets above, the word "precisely" marks the introduced condition/set as exhaustive—that is, a complete characterisation of the points within the parallelpiped—and thus isn't optional.