Difference between $\land$ and braces I was wondering what are the difference between the $\land$ and  $\begin{cases}
 \\
  \\
\end{cases}$ symbol. As I know, they both mean "and". So far, I've noticed the $\land $ on statements (not sure if it's the correct word) like : $$\forall x\in\Bbb R, \exp(x)\in\Bbb R\land \exp(x)>0$$
But I've seen braces on only multiple equation systems like :
$$\begin{cases}
x+y=0 \\
x-y=1  \\
\end{cases}$$
or in function definitions: $$|x|=\begin{cases}
-x,  & x<0 \\
x, & x>0  \\
\end{cases}$$
But only this difference seems too weak to have two different notations.
 A: I've never really seen "$∧$" being used the way you use it; it's usually preferable to write "and" in plain old English (of course, the symbol does come up in Boolean algebras and logic-related fields). For instance, usually you'll see something like

We have $\exp(x) \in \mathbb{R}$ and $\exp(x) > 0$ for all $x \in \mathbb{R}$

rather than 

$\forall x\in\Bbb R, \exp(x)\in\Bbb R\land \exp(x)>0$

The former is just easier to read. Braces are actually useful though, as they can be used to break up an expression into smaller parts. It's a convenient notation.
A: In your first example, with a system of equations, the brace is not essential, merely a comment on the equations to suggest that they are grouped together.  In your second example, a piecewise-defined function, the brace seems easier to understand than $$x<0\rightarrow |x|=-x \wedge x\ge 0\rightarrow |x|=x$$
(and if there are more than 2 pieces, the wedge notation becomes even more unwieldy).
A: Notation, much like language, is intended to clarify, not necessarily to express every idea as efficiently as possible.  It is just plain ugly to write a system as $(x,y) : (x+y = 0) \wedge (x-y=1)$.  Moreover, it is not logically correct to join non-Boolean expressions by $\wedge$, so I would find it very difficult to express the piecewise definition of $|x|$ using logical connectives.
