2
$\begingroup$

A partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

For example if $N=\{1,2,3\}$ then all possible partitions that can be formed are: $p_1=\{(1),(2),(3)\}$, $p_2=\{(1,2),(3)\}$, $p_3=\{(1,3),(2)\}$, $p_4=\{(2,3),(1)\}$ and $p_5=\{(1,2,3)\}$.

Does a definition exist, which is similar to the partition one but where elements are now allowed to be in multiple sets, a feasible partition of such definition would be $p=\{(1,2),(3),(1,2,3)\}$. That is, if $N$ is a set of players then players 1 and 2 belong in the same coalition with respect to some agreement $x$, 3 acts alone with respect to the same agreement, but with respect to some other agreement $y$ all players cooperate.

$\endgroup$
1
  • $\begingroup$ You should use the common notation $\{\}$ for sets. $(1,2)$ is usually used to denote an ordered pair which is not a set (the order of elements in a set does not matter). So for example you should write $p_2=\{\,\{1,2\},\{3\}\,\}$. $\endgroup$ – user477602 Oct 2 '17 at 23:15
3
$\begingroup$

A cover for a set $S$ is a set $X$ of distinct, non-empty subsets of $S$ such that $$ \bigcup_{}^{}X = S.$$

A partition for a set $S$ is a set $X$ of pairwise disjoint, non-empty subsets of $S$ such that $$ \bigcup_{}^{}X = S.$$

In other words, a partition of $S$ is a pairwise disjoint cover of S.

$\endgroup$

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.