# Existence of a definition similar to the partition

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.

• 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 $p_2=\{\,\{1,2\},\{3\}\,\}$. – user477602 Oct 2 '17 at 23:15

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.