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Simple question, is there any difference between $x\in (1,5)$, $x\in [1,5]$ and $x\in (1,5]$? and what is the value of $x$ in those cases?

Thank you.

Asaf Karagila
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ryan chandra
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  • Do you know what interval notation is? Each of those means a different sort of set. – Sean Roberson Dec 21 '22 at 03:14
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    Do you think $1\in (1,5)$? How about $1\in [1,5]$? – Gary Dec 21 '22 at 03:16
  • "What is the value of $x$ in each of those cases?" It is unspecified beyond what has already been said with the statement in question. $x\in[1,5]$ means that $x$ is a real number such that it is greater than or equal to $1$ and less than or equal to $5$. The statement would be true in the case that $x=2$ for instance, and would also be true in the case that $x=\pi$ or $x=4.7$ or even $x=5$ among infinitely many other possible scenarios. – JMoravitz Dec 21 '22 at 03:17
  • A set such as $(1,5)$ which does not contain its boundary points is also called open; a set such as $[1,5]$ which does is called closed. There is a good reason why the distinct notations exist: how would you denote the set $(1,5)$ using square brackets? $[1.01,4.99]$? $[1.000001, 4.999999]$? You can see how no such way of describing the set that starts "just after 1" and ends "just before 5" using square brackets would be adequate. – indnwkybrd Dec 21 '22 at 05:09

1 Answers1

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We have $$[1,5] = \{ x\in \mathbb R \mid 1 \leq x \leq 5 \},$$ $$(1,5) = \{ x\in \mathbb R \mid 1 < x < 5 \},$$ $$(1,5] = \{ x\in \mathbb R \mid 1 < x \leq 5 \},$$ so they all are sets that include all real numbers strictly between $1$ and $5$, but the first one also contains both $1$ and $5$, the second contains neither $1$ nor $5$ and the third one contains $5$, but not $1$.

When we write $x\in [1,5]$, we don't assign a particular value to $x$, we just say that $x$ is a real number such that $1 \leq x \leq 5$, and similarly for the others.

Ennar
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