Definition 6

Two sets A and B are said to be disjoint (or not to intersect) if $A \cap B = \emptyset$. If $A \cap B \neq \emptyset$, we say that A meets (or intersects) B. Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets. The sets of this family are said to be mutually disjoint if the conditions $\iota \in \mathrm{I}$, $\varkappa \in \mathrm{I}$, $\iota \neq \varkappa$ imply $\mathrm{X}_\iota \cap \mathrm{X}_\varkappa = \emptyset$.

Let $f$ be a mapping of A into B, and let $(\mathrm{Y}_\iota)_{\iota \in \mathrm{I}}$ be a family of mutually disjoint subsets of B. Proposition 4 then shows that the sets of the family $(\overset{-1}{f}\langle \mathrm{Y}_\iota \rangle)_{\iota \in \mathrm{I}}$ of subsets of A are mutually disjoint. On the other hand, if $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ is a family of mutually disjoint subsets of A, the sets of the family $(f \langle \mathrm{X}_\iota \rangle)_{\iota \in \mathrm{I}}$ are not in general mutually disjoint.