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I'm working through the basic math course on brilliant.org and I found an answer I don't understand here. (The 2nd part of the 4th answer option).

What does this mean?

-10 < a < 0

Is there an implicit AND after a ? And therefore does it mean this?

-10 < a AND -10 < 0
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    $\begingroup$ It means $-10 < a$ and $a < 0$. $\endgroup$ Aug 6, 2017 at 0:49
  • $\begingroup$ So there IS an implied AND in there. Great thank you! $\endgroup$ Aug 6, 2017 at 0:54
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    $\begingroup$ Bonus fact: sometimes mathematicians will use two less-than symbols to mean "much smaller than", as in $$\epsilon \ll 1$$ $\endgroup$ Aug 6, 2017 at 1:10

3 Answers 3

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There is an implied AND, but not in the way you put it. It really means $$-10 < a\quad\text{AND} \quad a<0 $$

In other words, $a$ is between $-10$ and $0$, excluding end points. In general, $x_1 < x_2 < x_3 <... < x_n$ means $$x_1 < x_2\quad\text{AND} \quad x_2 < x_3 \quad\text{AND} \quad x_3 < x_4 \quad ...\quad x_{n-2} < x_{n-1} \quad\text{AND} \quad x_{n-1} <x_n $$

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    $\begingroup$ Presumably, you mean "but not in the way you put it." $\endgroup$ Aug 6, 2017 at 0:59
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    $\begingroup$ @ThomasAndrews Yes, thanks. Must have typed a bit too quickly. Have corrected it. $\endgroup$
    – Eff
    Aug 6, 2017 at 1:25
  • $\begingroup$ What do you mean by excluding end points ? $\endgroup$ Aug 6, 2017 at 1:44
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    $\begingroup$ @AgentZebra I mean that $a$ is between $-10$ and $0$ but not equal to either $-10$ or $0$. Conversely, $-10 \leq a \leq 0$ would mean that $a$ is between $-10$ and $0$ including end points. $\endgroup$
    – Eff
    Aug 6, 2017 at 7:49
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it means that $a$ is between $-10$ and $0$

so $a<0$ and $a>-10$

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There is an implicit AND but not the way you think.

$-10 < a < 0$ means $-10 < a$ AND $a < 0.$

It is a convenient way to express the values a can take given an upper and a lower limit.

$\{x \mid -10 < a < 0\}$ is equivalent to $(-10, 0)$ in interval notation.

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  • $\begingroup$ Thanks, this helped me understand it. $\endgroup$ Aug 6, 2017 at 1:04

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