# What do two 'greater than' symbols mean

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

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

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 $$x_1 < x_2\quad\text{AND} \quad x_2 < x_3 \quad\text{AND} \quad x_3 < x_4 \quad ...\quad x_{n-2} < x_{n-1} \quad\text{AND} \quad x_{n-1} <x_n$$

• Presumably, you mean "but not in the way you put it." Aug 6, 2017 at 0:59
• @ThomasAndrews Yes, thanks. Must have typed a bit too quickly. Have corrected it.
– Eff
Aug 6, 2017 at 1:25
• What do you mean by excluding end points ? Aug 6, 2017 at 1:44
• @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.
– Eff
Aug 6, 2017 at 7:49

it means that $a$ is between $-10$ and $0$

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

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.

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