If I write $$ x\in [0,1] \tag 1 $$ does it mean $x$ could be ANY number between $0$ and $1$?

Is it correct to call $[0,1]$ a set? Or should I instead write $\{[0,1]\}$?


If I instead have $$ x\in \{0,1\} \tag 2 $$ does it mean $x$ could be only $0$ OR $1$?

  • $\begingroup$ Yes to all your questions. For the first one, it could be any number between $0$ and $1$ inclusive. Both $[0,1]$ and $\{[0,1]\}$ are sets. $\endgroup$ – Shirish Kulhari May 31 '18 at 11:57
  • $\begingroup$ Note that if you write $x \in [0,1]$ that means $0 \leq x \leq 1$, while $x \in (0,1)$ would be $0<x<1$, thus not including elements {0,1}. $\endgroup$ – A.T May 31 '18 at 12:15

If I write $x\in[0,1]$ does it mean that $x$ can be ANY number between $0$ and $1$?


If $x\in [0,1]$ then $x$ can be any number between $0$ and $1$ (inclusive). Another way to write this is $0 \leq x \leq 1$.

A related notation is $(0,1)$, or sometimes in European writing $]0,1[$, which is the open interval excluding end points, i.e. $0<x < 1$.

Is it correct to call $[0,1]$ as set? Or should I instead write $\{[0,1]\}$?

Yes, $[0,1]$ is a set (it is also called an interval because it contains only consecutive numbers). The set is $[0,1] = \{x\in\mathbb R\mid 0 \leq x \leq 1\}.$

However, $\{[0,1]\}$ is also a set. A different set. They are different sets because $[0,1]$ has an infinite (uncountable) number of elements (i.e. any real number between $0$ and $1$), whereas $\{[0,1]\}$ has only one element, namely $[0,1]\in\{[0,1]\}$.

If I instead have $x\in\{0,1\}$ does it mean that $x$ could be only $0$ OR $1$?


If $x\in\{0,1\}$ then $x$ is either $0$ or $x$ is $1$, and not for example $0.312$.

  • $\begingroup$ Great, thanks! So with $[0,1]$ we mean the set: $$[0,1] = \{x\in\mathbb R\mid 0 \leq x \leq 1\}$$ does it mean $\{0,1\}$ is the set $$\{0,1\}=\{x\in \mathbb R\mid x=0 \lor x=1\}\quad ?$$ $\endgroup$ – JDoeDoe Jun 2 '18 at 9:09
  • $\begingroup$ @JDoeDoe Yes, that is correct. $\endgroup$ – Eff Jun 2 '18 at 12:15

$[0,1]$ is (defined as) the set $\{ x \in \mathbb{R} : 0 \leq x \leq 1 \}$, i.e. it is a set that contains every real number between $0$ and $1$ (inclusive). It contains an uncountable number of elements.

$\{0,1\}$ is a set containing 2 elements: $0$, and $1$.


Is it correct to call $[0,1]$ a set?

Yes, although it doesn't sound natural to me if you "call $[0,1]$ a set". I'd rather call it the closed interval. You can also write as $\{x|x\in[0,1]\}$ (trivial).

$x\in \{0,1\}$ means $x=1$ or $x=0$.

  • 3
    $\begingroup$ I think it is common to consider intervals as (particular type of) sets. $\endgroup$ – Surb May 31 '18 at 12:18
  • $\begingroup$ I'm being subjective here. Personally I don't "call" $[0,1]$ a set (if I have to, I'd avoid brackets), but it IS a set. If you call it a set, that's of course correct in mathematics. $\endgroup$ – poyea May 31 '18 at 12:27

$x\in[0;1]$ means $0\le x\le 1$, any real number $x$ that satisfies this is true. For example:

$x^2-x\le 0\Leftrightarrow x(x-1)\le 0\Leftrightarrow x-1\le 0\le x\Leftrightarrow 0\le x\le 1$

$x\in\{0;1\}$ means $x=0$ or $x=1$ is true. For example:

$x^2-x=0\Leftrightarrow x(x-1)=0 \Leftrightarrow x\in\{0;1\}.$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.