Notation for the set $\{0,1\}$

Question

When doing some complexity theory, I get bored of typing all the time the set $\{0,1\}$. Is there some widely used alternate fancy notation?

Answer 1

How about using $2$ or maybe $\mathbf{2}$ if you also use the number $2$ in other contexts and want to avoid confusion? In fact, under a standard construction of the natural numbers in set theory we indeed have the equality $2 = \{ 0, 1 \}$ (see here) and this notation is quite common in set theory and topology.

Answer 2

I have seen $\mathbb Z_2$ used to denote that set, with obvious extension to $\mathbb Z_n $

Answer 3

Adding another notation. If you have used Finite fields then GF(2) or just $\text{F}_2$ must suffice.

Answer 4

This is denoted by $\mathbb{F}_2$, the simplest field. Note that in advanced math, $\mathbb{Z}_2$ uses for dyadic integers which is an uncountable set.

Answer 5

In short: there may be, but you should not.

If you are doing research in complexity theory, your goal is not only to derive new results and ideas, but also, crucially, to communicate them to others. In particular, talking a language the other researchers in that field are familiar with is paramount: nobody wants to decipher a paper where the author chooses, for no apparent purpose, different notations for every single notion. So if your rationale is that

[you] get bored of typing all the time the set $\{0,1\}$.

and are looking for

alternate fancy notation

(emphasis mine), then the answer is: don't do it. Using a "widely used" notation from a different field may sound cool, but if the complexity theorists don't know it, then it's a bad idea. And if you have to ask for alternate notations, it means you haven't seen them in the papers of complexity theory that (I assume) you have read, so... they are not, by definition widely used in complexity theory.

Now, don't get me wrong:

• people in (algebraic) property testing and some subfields complexity theory often use $\mathbb{F}_2$; but it's because they want to use the algebraic structure of $\mathbb{F}_2$, and often consider $\mathbb{F}_p$ as well;

• people in Boolean Fourier analysis will often switch to $\{-1,+1\}$ (actually, $\{+1,-1\}$), but because it is convenient (to be able to write the characters $\chi_S$ in a multiplicative way, which often simplifies the proofs and derivations).

In both cases/examples above, people do this for a reason, and it's widely recognized in these respective areas of (or intersections of these areas with) complexity theory. Boredom is not such a reason.