Is there a difference between a range and an interval?
Can the terms 'interval' and 'range' be used interchangeably or do they describe different things?
I am talking specifically about sets of numbers under a suitable $<$ relation, such that they can be described as $r = [a,b]$, $(a,b)$, $[a,b)$, or $(a,b]$, meaning that for any $a < c < b$, we have $c \in r$ (and $a$ and $b$ depending on the respective brace used).
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$\begingroup$Can be. But convention gives clarity by usage/familiarity.
If $y=f(x),$ then you can consider an interval anywhere in the full extent of domain of independent variable $x$. Range is used for difference of max/min of dependent variable $y.$
$\endgroup$ $\begingroup$Your sets above are called intervals. "Range" is unusual .
$\endgroup$ 2 $\begingroup$'interval' versus 'range': interchangeable? or describe different things?
First, from the point of view of modern mathematics.
The function idea, which carries with it the domain, co-domain, and range concepts, is utterly fundamental. So when saying "range" to a mathematician, expect that the range of some given function will be the first thing that will come to mind. Our mathematician will not presume that this range is an interval, being accustomed to exotic functions taking values in arbitrary sets. (I take interval to mean a set $r$ as described in your question, although we might also allow intervals with only one end-point, e.g. $r=\{c:c<b\}.)$
But we should right away note, as we pass from professional mathematicians to the wider class of people who have taken (been subjected to?) calculus or "pre-calculus" algebra (USA jargon), that a great many of the functions studied have ranges that happen to be intervals or unions of intervals. Conversely, given an interval such as $[1,3)$, it is no trouble to construct several different continuous functions having this interval as its range.
So when talking to such people (say a randomly selected programmer), we must admit that the case for interchangable has some support. (Perhaps since coursework tends to deal with continuous functions.) This is further supported by a glance at the encyclopedia and the dictionary:
Interval (mathematics), also called range, a set of real numbers that includes all numbers between any two numbers in the set (from )
Range (noun) (1.) = The area of variation between upper and lower limits on a particular scale, e.g. "the cost will be in the range of USD 1–5 million a day" (from lexico.com)
Range (verb) (1.) = Vary or extend between specified limits, e.g. "patients whose ages ranged from 13 to 25 years" (from lexico.com)
I have found that in conversation with laymen, on ordinary topics, the word "range" is more commonly heard. But again, this is a mathematics site, and so from the point of view of English used for mathematics, the answer I think must be "describe different concepts", but with plentiful instances in which an interval happens to be the range of some $f$, and instances in which some given function maps its domain onto some $r$.
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