# Notation for a function with the domain $1$ or $0$?

I don't know if "binary function" is the correct name for the following?

I have the functions $f$, $g$, and their domains are only $1$ or $0$.

What is the correct notation for this?

I have the following (? are $1$ or $0$): \begin{align} f:?\rightarrow \mathbb R \tag 1 \end{align}

And for the vector-valued function $g$: \begin{align} g:?\rightarrow \mathbb R^n \tag 2 \end{align} where $g(?)=(g_1(?), \dots, g_n(?))$.

Thanks!

• To clarify, the $?$ in your example is the set $\{0,1\}$? – Patrick Stevens Feb 6 '18 at 9:14
• Hi @PatrickStevens! I guess it is the correct term. I mean $f$ could only take the values $f(1)$ or $f(0)$ and the same for $g$. – JDoeDoe Feb 6 '18 at 9:16
• That's again not quite the same thing, but I think you mean that $f$'s domain is $\{0,1\}$ :P What you just said in your comment is equivalent to the range of $f$ being of size at most $2$. – Patrick Stevens Feb 6 '18 at 9:18
• Hi @PatrickStevens Ah, my fault. I mean the domain/input for the function. The range is $\mathbb R$ for $f$. – JDoeDoe Feb 6 '18 at 9:24
• A pseudo-boolean function appears to be almost what you want. en.m.wikipedia.org/wiki/Pseudo-Boolean_function?wprov=sfla1 – Mark Feb 6 '18 at 9:30

I do not think that the term "binary function" would be appropriate. Indeed, since a binary sequence is a function from $\mathbb{N}$ to $\{0, 1\}$, I would rather see a binary function as a function with range $\{0, 1\}$.

I have no good suggestion for a better name, but you could represent such a function as a pair of reals $(r_0, r_1)$ (where $f(0) = r_0$ and $f(1) = r_1$).

These are similar to two-element tuples, with indexes taken from $\{0,1\}$. The elements are reals or vectors.

Such tuples can be called doubles, couples, pairs or duads. (https://en.wikipedia.org/wiki/Tuple#Names_for_tuples_of_specific_lengths)

Then I would suggest duad functions.

• In US English, I've never heard/read duad, and wouldn't be sure what you meant if you had said "dyad" functions either. – Mark S. Feb 6 '18 at 10:58
• @MarkS.: this is not US English, this is maths English. Check the Wikipedia entry. – Yves Daoust Feb 6 '18 at 11:01
• What I mean to say is that I think this is rare enough to not be a great idea. Monad, triad, tuple, and pair are all relatively common in comparison. Even Wikipedia disagrees on how the greek duad should be used as en.wiktionary.org/wiki/duad suggests it's used in math for unordered pairs (what I've heard called a "doubleton" occasionally). – Mark S. Feb 6 '18 at 12:07