Collections of values
Sets are used to represent collections of distinct values (referred to as members ot elements). Multiple sets, or sets of pairs can sometimes be represented as an adjacency matrix.
There are two ways in which a set can be specified, as either a list or a generator.
A list specification for a set would simply be written as $bb “S” = {1,2,3,4}$, where the members of the set are given as a comma separated list within braces. It is convention that a value representing a set is bold.
A generator specifies the members of the set based on a logic statement. The list given above would be represented as $bb “S” = {x | 0 < x < 5}}$, where the value given before the bar is the member, with the values being all of those which satisfy the logic statement after the bar.
An element being a member of a set is expressed as $1 in bb “S”$.
There are a few common sets that are referred to by standard symbols:
A subset is a set where its members also all belong to another set, so $bb “X” sube bb “Y”$ when X contains only elements from Y, but not necessarily all of them. It is more precisely a subset iff every member of X is a member of Y.
A proper subset, expressed by ⊂, holds if it is a subset but not equal to the other set. More simply, a proper subset has at least one fewer element than the parent set.
The intersection of two sets, represented by ∩, is the set which contains only the values which are members of both sets. This if formally given as $bb “X” nn bb “Y” = {x | x in bb “X” ^^ x in bb “Y”}$.
Intersection is a commutative operation.
The union of two sets, represented by ∪, is the set containing the elements of both sets. Since a set must have distinct values, each value will only appear in the set at most once.
There is also the notion of a disjoint union, given as ⊎, where the two sets have no members in common. This is, in practice, no different to a union operation.
Union is a commutative operation.
The difference of two sets, represented as a backslash, is those elements which feature in the first but not the second. Therefore, $bb “X” \\ bb “Y” = {x | x in bb “X” ^^ x !in bb “Y”}$.
The complement of a set, represented as X’, is the set which contains all elements other than those in the original set. This is the difference of the universe and the original set.
The cardinality is the count of members in the set. Cardinality is often represented in the same way as magnitude would be, |N|, or with a hash, #N.
The powerset of a set S is given as $2^(bb “S”)$. It contains all possible subsets of the set, including the set itself and the empty set.
The cartesian product of two sets, given by x, is a set containing orders pairs of each combination of elements. If there were two sets $bb “A” = {1,2,3}$ and $bb “B” = {4,5,6}$, then $bb “A” xx bb “B” = {(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}$.