# Why does the interval [1,1) not contain 1?

On a quiz I was asked whether the interval [1,1) contained the number 1, and I answered that it was true, but apparently the answer is false.

But why doesn't it? I thought that "[1,1)" means "all numbers from 1 inclusive and 1 exclusive", so since 1 is included at the start it doesn't matter if it's excluded at the end since we already included it...

• Usually $[a,b) := \{x\in\mathbb{R}:\ a \leq x < b\}$. – Rigel Oct 19 '18 at 16:25
• $[1,1)=\{x : 1 \leq x <1\}=\emptyset$ – Chinnapparaj R Oct 19 '18 at 16:26
• I think, @ChinnapparajR, you mean \emptyset or \varnothing ($\emptyset$, $\varnothing$), and not $\phi$. – Namaste Oct 19 '18 at 16:28
• Think of it this way: "$[a,b)$" means "The set of $c$ which are $\ge a$ and $<b$. It's that "and" which causes the problem. – Noah Schweber Oct 19 '18 at 16:28
• @amWhy Sadly, some texts do use $\phi$ for the emptyset (I personally hate this, but it is a usage that occurs). – Noah Schweber Oct 19 '18 at 16:29

Hint: $$[a,b) = \{ x \in \mathbb R : a \le x < b \}$$. What real numbers $$x$$ satisfy $$1 \le x < 1$$ ?

[a,a] = {a}

The only real number $$x$$ where $$a\leq x\leq a$$ is $$a$$.

[a,a) = $$\phi$$ (no items)

There is no real number $$x$$ where $$a\leq x.

(a,a] = $$\phi$$ (no items)

There is no real number $$x$$ where $$a< x\leq a$$.

(a,a) = $$\phi$$ (no items)

There is no real number $$x$$ where $$a< x< a$$.

No need to think more. HintWhat do you say about 0≤x<0. Is there exist any real number? Think about it. You will get it.