Notation for possible values of a random variable Let $X$ be a discrete random variable that can take the values $1, 2, \textrm{and}\ 3$. What is the conventional way to write this mathematically? Is it just $X\in\{1,2,3\}$ or should I somehow write the pmf.
 A: There are sloppy ways and rigorous ways.  The sloppy ways are shorthands, like "$X\in\{1,2,3\}$", that are either nonsensical or (in this example) just plain wrong when interpreted according to the correct conventional meanings of the symbols.  (The second statement literally means $X$ is one of three specified integers--which aren't random variables at all.)  Such a shorthand can be effective in contexts where (a) its meaning is defined and (b) set-theoretic notation will not otherwise be used.
Recalling that all random variables are measurable functions defined on probability spaces, a standard way to stipulate that $X$ can take on a given set of values is to use functional notation to specify its image, as in
$$X:\Omega\to\{1,2,3\}$$ 
or
$$X(\Omega) = \{1,2,3\}.$$
Many statistical writers eschew such notation because they prefer to suppress all references to $\Omega$, which is intended to remain abstract, or they even think of random variables as being some kind of class of objects in which $\Omega$ is not even a definite set.  An equivalent notation avoids referencing $\Omega$, such as stipulating
$$\text{Image}(X) = \{1,2,3\}.$$
Very confusingly, some statistical writers use the term "domain" to refer to the image of $X$ (whereas in mathematics the word "domain" invariably refers to $\Omega$!).  A Google search will easily turn up such uses of "domain."  Others use phrases like "defined on" or "take on," such as "$X$ takes on the values $\{1,2,3\}$" or "$X$ is defined on $\{1,2,3\}$" (by which they really mean the probability mass function of $X$ rather than $X$ itself).
There are indirect ways to refer to the image of $X$.  For instance, real-valued random variables are often thought of as being almost interchangeable with their distribution functions.  The support of such a function has a well-established definition in probability and measure theory; in the case of a finite discrete random variable $X$, it will coincide with the image of $X$.  People who write about such things typically adopt some mnemonic notation for this, such as
$$\text{supp}(X) = \{1,2,3\}.$$
Finally, this helps us understand how such confusion about the meaning of "domain" can arise.  If we were to conflate the random variable $X$ with its probability mass function (pmf) $p_X$, given by the probabilities
$$p_X(x) = \Pr(X=x),$$
then in the example $p_X(x)\ne 0$ only when $x \in \{1,2,3\}$.  We could, if we wished, restrict $p_X$ (which notionally is a function defined on $\mathbb{R}$) to the subset $\{1,2,3\}\subset\mathbb{R}$ without losing any of the information it conveys.  This would make $\{1,2,3\}$ the (mathematically correct) domain of the restricted $p_X$.
A: A full description of $X$ would include the pmf, yes.  For example ...
$$
X \in \{1,2,3\} \\
\mathbb{P}(X=1)=\frac{1}{12} \\
\mathbb{P}(X=2)=\frac{7}{12} \\
\mathbb{P}(X=2)=\frac{1}{3}
$$
... which looks a bit clumsy, or do it in words: $X$ is a discrete random variable in $\{1,2,3\}$ with probabilities $p_1=\frac{1}{12}, p_2=\frac{7}{12}, p_3=\frac{1}{3}$ respectively.
