Example

For every non-empty set A, the family $(\{x\})_{x \in \mathrm{A}}$ is a partition of A.

If $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ is a partition of a set E, the mapping $\iota \rightarrow \mathrm{X}_\iota$ of I onto the set $\mathfrak{F}$ of elements $\mathrm{X}_\iota$ of the partition is bijective. Hence, if $\mathfrak{F}$ is given, the partition is determined up to a one-to-one correspondence between index sets. Usually when we speak of a partition, it is the set of elements of the partition with which we are concerned.