XS. Sets

Sets can be defined by some rule, or by listing members. Both approaches can be handled through synthetic notation. The most commonly designated sets, such as the set of all prime numbers, can also be handled synthetically, though it will often not be necessary to utilize the term “set.” Disjoint sets are sets that do not intersect. These too can be represented synthetically.

1. Groups. These are a type of set for which there is at least one mathematical operation that connects group members (as addition of any two integers generates another integer). Again, particular groups can be identified synthetically.

a. Rings. A type of group exhibiting additional characteristics.

aa. Fields. A type of ring exhibiting additional characteristics.

2. Classes. These are defined differently than ‘sets’ within contemporary set theory, though the terms have often been used synonymously in the past. As with sets, particular classes can be identified synthetically. Though not a subset of ‘sets’ it may be advisable to place them here to indicate their similarity.