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Let us define the set $[a,b) = \{ x \in \mathbb{R}: a\le x <b\}$

Is $[a, a)$ equal to $\{a\}$ or $\varnothing$?

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    $\begingroup$ A logician would tell you that the answer is "yes". $\endgroup$
    – beep-boop
    Commented Oct 15, 2014 at 22:17

4 Answers 4

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You can write the conditions as follows: $[a,a)=\left\{x\in\mathbb{R}: a\leq x<a\right\}$. Now, since there is no real number fulfilling this condition (which, if fulfilled, implies $a<a$), the set must be empty.

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It equals the empty set, while $[a,a] = \{a\}$.

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  • $\begingroup$ I have clarified the question. Please review your answer. $\endgroup$ Commented Oct 15, 2014 at 21:57
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    $\begingroup$ I think it remains: $[a,a) = \varnothing$, just clarifying that it's $[a,a]$ that would give you $\{a\}$. $\endgroup$
    – Platehead
    Commented Oct 15, 2014 at 21:59
  • $\begingroup$ That's fine. I just didn't want to give you a nasty surprise. $\endgroup$ Commented Oct 15, 2014 at 22:00
  • $\begingroup$ I think this is more of a philosophical question. It can be both, it is like, "glass half empty or half full" type question. @Platehead here is a pessimist...:P $\endgroup$ Commented Oct 15, 2014 at 22:02
  • $\begingroup$ Aww really? It seems to just make sense according to the definition: nothing is both $\geq a$ and $< a$ right? :) $\endgroup$
    – Platehead
    Commented Oct 15, 2014 at 22:03
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You can think of $[a,b) = (-\infty,b) \cap [a,\infty)$. Choose $b=a$ :D

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From Wikipedia:

  • $[a, a) = (a, a] = (a, a) = \emptyset$ is the empty set.
  • $[a,a] = \{a\}$ is a degenerate set.
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