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I am writing an exercise for a precalculus homework assignment that deals with the topic of interval notation. The point of the exercise is to convert open, closed, and half open intervals described in English into interval notation. Here is a snippet of the exercise:

Write the following sets of numbers represented textually in interval notation.

(a) All real numbers between 5 and 7 that include 5 but don't include 7.

(b) All real numbers between 1 and 10 that include both 1 and 10.

So for example, the answer to part (a) would be $[5, 7)$.

I feel as though my wording is awkward and confusing in both the prompt and parts (a) and (b). Could anyone give me advice on re-wording or re-imagining this question to make it more clear?

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    $\begingroup$ (a) says that the numbers have to include 5 and not include 7? So 5.5 is okay, 5.55 is okay, 5.57 is not okay, 5.6 is okay, ... $\endgroup$
    – user253751
    Aug 28, 2020 at 13:20
  • $\begingroup$ @user253751 if I wanted to say that I would probably say "numbers that do not include a 7" or even "numbers that do not include the digit 7in their decimal representation." $\endgroup$
    – barbecue
    Aug 28, 2020 at 16:09
  • $\begingroup$ @user253751: If we're going to go down this path, then the fact that "numbers" is used and not "numerals" addresses your concern. $\endgroup$
    – Dave L Renfro
    Aug 28, 2020 at 16:50
  • $\begingroup$ @user253751 I think he means greater than or equal to 5 and less than 7. 5 <= x < 7. $\endgroup$
    – Bloke Down The Pub
    Aug 29, 2021 at 0:20
  • $\begingroup$ If the students are up to it, you might want to include some exercises with decimals or fractions, lest you build up a misunderstanding that e.g. [5, 7[ = {5, 6}. $\endgroup$
    – Tommi
    Mar 10 at 8:20

4 Answers 4

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This is my suggestion.

Write each set of numbers in interval notation.

(a) All real numbers between 5 and 7, including 5 but not including 7.

(b) All real numbers between 1 and 10, including both 1 and 10.

I assume you are giving examples in class before you assign this.

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If you feel your instruction is unclear, then provide an example.

Write the following sets of numbers represented in interval notation. For example, "all real numbers from 0 to 1 exclusive" in interval notation is $(0,1)$.

a) All real numbers greater than or equal to 5 and less than 7.

b) All real numbers from 1 to 10 inclusive.

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  • $\begingroup$ I generally like Amy B's suggestion, but +1 for suggesting alternate phrasings. I assume the actual exercise will include more than two phrases, in which case illustrating different ways of specifying similar intervals can be useful. $\endgroup$
    – Ilmari Karonen
    Aug 28, 2020 at 14:41
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    $\begingroup$ Isn't the entire point to check that the student knows that $($ is exclusive? $\endgroup$
    – chrylis -cautiouslyoptimistic-
    Aug 28, 2020 at 16:08
  • $\begingroup$ @chrylis-cautiouslyoptimistic-, could you please clarify? Are you saying that the example provided gives away the answer? $\endgroup$
    – JRN
    Aug 29, 2020 at 6:47
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Write the following sets of numbers represented textually in interval notation --- Note that this can be parsed in two different ways: "Write the following sets of numbers (represented textually in interval notation)" and "Write the following sets of numbers represented textually (in interval notation)". The first asks for sets of numbers to be written, where the sets of numbers are given in a textually represented form that is written in interval notation. The second asks for sets of numbers to be written in interval notation, where the sets of numbers are given in a textually represented form. Of course, context prevents one from parsing this in the first way, but students who are very uncertain about what this is all about could easily parse this in the first way and be confused as to what it is you want.

For each of the following textual descriptions of a set of numbers, write the set of numbers in interval notation.

(a) All real numbers between 5 and 7, including 5 and not including 7.

(b) All real numbers between 1 and 10, including 1 and including 10.

Note that I've written (a) and (b) in a similar fashion (this is called parallel writing), and there are 4 formats that the words after the comma can take: (1) including $m$ and including $n$; (2) including $m$ and not including $n$; (3) not including $m$ and including $n$; (4) not including $m$ and not including $n$. For an individual class, being this meticulous is not particularly important, but if these questions are to appear on a widely taken standardized test in which the results are to be analyzed statistically, the use of parallel writing tends to decrease irrelevant variance caused by differing reading skills and other factors that one would want to disentangle from the math skills being measured.

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I'm totally nit-picking, as requested.

Write the following sets of numbers represented textually in interval notation.

This instruction is problematic because it suggests that the given information is in interval notation; editing it to "Write, in interval notation,..." rectifies this issue, but the phrase "represented textually" still imposes unnecessary cognitive burden.

I suggest:

  • Use interval notation to denote the following sets:

(a) All real numbers between 5 and 7 that include 5 but don't include 7.

(b) All real numbers between 1 and 10 that include both 1 and 10.

"All objects that have property P..." refers to objects that each have property P, whereas "the set of objects that has property P..." refers to objects that collectively have property P:

your problem statement suggests that the number $5$ “includes” $5$ (and that $6$ “includes” $5);$ this is not entirely meaningful.

I suggest:

  • (a) the set of real numbers that are smaller than $7$ but not smaller than $5$
  • (a) the set of real numbers that are greater than or equal to $5$ and smaller than $7$
  • (a) the set of real numbers between $5$ to $7,$ including the former but not the latter

  • (b) the set of real numbers from $1$ to $10$
  • (b) the set of real numbers between $1$ to $10,$ inclusive
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