I would like to give a name to the form (say XYZ form) where a set is only represented as union of its subsets and the subsets are mutually disjoint and themselves only represented either as explicit enumeration of elements or in X form themselves. The question is, if there is already a well-known name for this form?
Essentially, if $U$ is a universal set. A set representation of a set $R$ of a set $X \subset U$ is said to be in XYZ form if
- $R$ has the form $\{a_1, a_2, \ldots, a_n\}$. (The elements are all listed explicitly)
- $X = Y \cup Z$ where $Y$ and $Z$ are in XYZ form.
Examples:
- $A = \{a\}$ is in XYZ form.
- $B = \{b,c\}$ is in XYZ form
- $A \cup B$ is in $XYZ$ form
- XYZ form of $A \times B$ would be $\{(a,b),(a,c)\}$ or it could be $\{(a,b) \} \cup \{(a,c)\}$
- Let the universal set $U$ be $\{a,b,c\}$, and $A=\{a\}$, and $B=\{b\}$ then the XYZ form of $(A \cup B)^c $ would be $\{c\}$.
- $A \cap B$ is not in XYZ form. The XYZ form of $A \cap B$ is $\{\}$ (the empty set)