Below is a question I encountered:


My answer was $\{x \mid x \in Z, -1 \leq x \leq 4 \}$

However, the textbook answer stated $\{x \mid x \in Z, -1 \leq x < 5\}$

Can $\leq$ and $<$ be used interchangeably in this situation? Is there a more accepted convention?



  • $\begingroup$ Both are correct if it's a question about integers. I don't know of a convention. I prefer the first one.because you have to read the second one particularly carefully to check it. $\endgroup$ – Ethan Bolker Mar 8 '17 at 0:12
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    $\begingroup$ I think the photo is cut off. I can't see the numbers on the bottom number line :| $\endgroup$ – Timothy Cho Mar 8 '17 at 0:12
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    $\begingroup$ Perhaps your book was written by a computer scientist :) [When describing a range of integers, many programming languages and many style conventions will include the left endpoint but exclude the right]. $\endgroup$ – benguin Mar 8 '17 at 0:21

I don't think that there's a convention about if you should use $\leq$ or $<$, but I do think that whichever you do you should be consistent. Writing $-1\leq x<5$ like the book has can very easily lead to confusion because people will miss the fact that the "or equal" was dropped. In fact, when I read this at first I went "Ah ha! The book has a typo, it should say $-2<x<5$."

  • $\begingroup$ In computer science, it's generally considered better to write $−1\leq x<5$ or $x\in[-1, 5)$, and use half-open intervals generally. Why? Because the algebra of half-open intervals is nicer: the length of the interval $[a, b)$ is $b-a$, not $b-a-1$; if we split $[a, c)$ at $b$, we get $[a, b)$ and $[b, c)$ instead of $[a, b-1]$ and $[b, c)$, etc. In general, we avoid double counting the boundaries, or having to sprinkle "-1" all over the place. $\endgroup$ – Max Apr 13 '17 at 7:54

If we take you literally, the answer to your question "Can $\leq$ and $<$ be used interchangeably in this situation?" is no: $\leq$ and $<$ cannot "be used interchangeably." For the statement $x\leq 4$ is not equivalent to the statement $x<4$. You can't interchange $\leq$ with $<$ without changing something else in the statement.

But perhaps you didn't write what you meant to ask. Perhaps you meant: is there a reason to prefer the notation $x\leq 4$ to the notation $x<5$ when $x$ denotes an integer? In that case, the answer to your question is still no: it's just a convention; both inequalities refer to the same set of integers.

  • $\begingroup$ Sorry about the confusion, yes I meant the question in the latter paragraph. $\endgroup$ – George Tian Mar 8 '17 at 2:39

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