Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a set $S = \{1,2,\ldots,n\}$ of $n$ elements and I denote with $P(S)$ the powerset of $S$.

Which is a correct and accepted notation to say that the set $Z$ is composed by all the elements in $P(S)$ with the exception of all the subsets of cardinality $h$?

Example: if $S=\{1,2,3\}$ then $P(S) = \{\emptyset, \{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$.

If $h = 1$, then it should be $Z = \{\emptyset, \{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}=P(S)\setminus\{\{1\},\{2\},\{3\}\}$.

How can it be expressed in a formal and concise way, for any value of $n$ and $h$?


share|cite|improve this question
The notations $[X]^h$ and $X^{[h]}$ are both used to denote the collection of subsets of $X$ of size $h$, so one possibility is $\mathcal P(S)\setminus[S]^h$. I haven't seen any specific notation for this set, though $[S]^{\ne h}$ would be reasonable. – Andrés E. Caicedo Aug 6 '12 at 23:04
I've also seen $\binom{X}{h}$ for $h$-subsets of $X$, but I can't find a reference (so it's obviously not standard). Another option is to take this further and write $\mathcal{P}_{\ne h}(S)$. (Also note $[n]=\{1,\cdots,n\}$ is fairly standard shorthand in analytic number theory and combinatorics.) – anon Aug 6 '12 at 23:06
up vote 2 down vote accepted

$$ Z = \left \{ x | x \in P(S) , |x| \neq h \right\} $$

An alternate, more compact version:

$$Z = \{x \in P(S) : |x| \neq h\}$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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