1
$\begingroup$

$2^n$, with $n$ corresponding to the number of elements, is an almost unavoidable calculation involved in counting. At the root of its usefulness there lies - I presume - the identity:

$$\sum_{0\leq{k}\leq{n}}\binom nk =2^n$$

that explains its application to the calculation of the power set or $2^S$ or the set of all subsets of $S$. However, it is also used without much need for mathematical theory in day to day calculations, such as determining the different ways that $+$'es and $-$'es can be assigned to a list of numbers, say from $1$ to $3$ (i.e. $n=3$): $1$ can be positive or negative ($2$), and for each branch, $2$ can be positive or negative ($2^2)$, and for each choice, $3$, in turn, can assume positive or negative signs, for a total of $2^3$ ways of allocating signs.

In looking up for a name akin to combinations for $\binom nk$, or permutations for $P(n,k)$ for this ubiquitous $2^n$ operation, I have found "number of $k$-combinations for all $k$, with an explanation as to its correspondence to the sum of the $n$-th row of Pascal triangle.

Very neat. But is there a mathematical name that directly and succinctly identifies this $2^n$ operation of counting a process for which each node or step splits into a binary choice?

$\endgroup$
  • $\begingroup$ $2^n$ is so simple to write that it doesn't need any other specific name or notation. $\endgroup$ – Henning Makholm Nov 10 '15 at 19:24
  • $\begingroup$ @Jean-ClaudeArbaut Thank you. I think you answered my question. I'd be happy to accept it if I could. $\endgroup$ – Antoni Parellada Nov 10 '15 at 19:26
  • $\begingroup$ If you wish :-) $\endgroup$ – Jean-Claude Arbaut Nov 10 '15 at 19:28
  • $\begingroup$ It is the number Of subsets Of a set with n elements. But this is not as succinct as hoped. $\endgroup$ – Jean-François Gagnon Nov 10 '15 at 19:29
1
$\begingroup$

I don't think there is a short name for $2^n$. However, I think that "the number of subsets of a set with $n$ elements" (or cardinal of the powerset, of course) is already nice enough. For instance, it fits very well with your exemple of +'es and -'es: just select the subset of positive terms.

$\endgroup$
1
$\begingroup$

I would call it the base $2$ exponential function. If this doesn't work for you, perhaps you could call it the doubling function?

$\endgroup$
1
$\begingroup$

$2^n$ is a special function not only for the reason specified. It is the discrete analog of the continuous function $e^x$. Namely, it is the unique nontrivial function whose first difference is itself. $$\Delta 2^n = 2^{n+1}-2^n=2\cdot 2^n -2^n=2^n$$ (while in the continuous case $e^x$ is the unique nontrivial function whose first derivative is itself)

$\endgroup$
  • $\begingroup$ This is very cool! Thank you. I was looking for a quick connecting name, much like when I think "permutations" I think "factorial" (n!); and when the problem is one of "combinations" I think "binomial". In trivial math problems like the one I explain (+ and -) the idea of "power sets" doesn't come to mind immediately... $\endgroup$ – Antoni Parellada Nov 10 '15 at 19:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.