# Theory of proveit.logic.sets¶

Basic set theory theory dealing with membership (e.g., $x \in S$), enumeration (e.g., $\{e_1, e_2, e_3\}$), containment (e.g., $A \subseteq B$), unification (e.g., $A \cup B \cup C$), intersection (e.g., $A \cap B \cap C$), subtraction (e.g., $A - B$), comprehension (e.g., $\{f(x)~|~Q(x)\}_{x \in S}$), and cardinality (e.g., $|S|$).

In [1]:
import proveit
%theory


### Local content of this theory

common expressions axioms theorems demonstrations

### Sub-theories

membership Is an element a member of a set? Not a member of a set? Do sets have the same elements? Define a set by enumerating its contents. Is one set included in another (as a subset)? Define a set as the union of sets. Define a set as the intersection of sets. Define a set by removing elements from an original set. Define a subset according to the properties of its members. Define a set as the power set of another set. Define a set as all combinations of elements of other sets. Are there common elements between sets? Also defines "distinct". How large is a set? A function is a mapping from a domain to a codomain (both sets) and has various properties.

### All axioms contained within this theory

This theory contains no axioms directly.