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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.

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Wenn you just write out function names like that, $\TeX$ interprets that as a juxtaposition of variable names and formats it accordingly. To get the appropriate font and spacing, you can use predefined commands like \exp, or, if you need an operator name for which there isn't a predefined command, you can use \operatorname{name}. There's an edit link underneath the question. – joriki May 4 '13 at 16:33
Thanks edited the formatting – moray95 May 4 '13 at 16:37
up vote 2 down vote accepted

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.

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Actually, I have the intuition that mixing math and english language is "ugly". Like I'd prefer to write "we have exp(x) in $\Bbb R$ and exp(x) greater than 0 for all x in $\Bbb R$" in plain english rather then what you wrote. – moray95 May 4 '13 at 16:56
@moray95 It depends on the context, and it's obviously a matter of preference. But speaking from experience, I'd say that what I wrote down is a fair representation of the style of writing you usually see in a math textbook. – Alex Provost May 4 '13 at 17:00

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).

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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.

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