I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, is it really? I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, and that I should always write:
$$
x\equiv a\pmod {d}\text{ or }x\equiv b\pmod {d}\text{ or }x\equiv c\pmod {d}
$$
How true is this?
 A: The acceptability of any abuse of notation depends on whether the meaning is clear. For instance if you are asked to solve the equation $x^2 - 3x + 2=0$ and write $x=1,2$ I think most everyone will know what you mean, although to be precise you should write $x = 1$ or $x=2$, or possibly even $x \in \{1,2\}$.
At first glance the meaning of $x \equiv a,b,c (\mathrm{mod}\, d)$ is not all that clear, but upon reflection it is just the same type of abuse of notation as above. As long as the meaning is clear it is okay, but bear in mind a nonstandard use of notation will probably be unclear to most people reading what you write.
A: I agree that it's bad notation to write it that way.  I would read what you wrote as asserting that all three statements below hold, so it's somewhat ambiguous and nonstandard at best.  Putting the word "or" in the equation as pjs36 suggests is better, but it's kind of strange to have a word in the middle of an expression like that.
You could write
$$(x \bmod d) \in \{a, b, c\}$$
if $a$, $b$ and $c$ have already been reduced modulo $d$, however.
