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> Sets
> Definitions
DefinitionsThe definitions of sets and related terms.Sibling topics:Contents:
Definition of a set
A set is an unordered collection of objects. The objects in a set are called its elements or
members.
Sets contain only distinct objects, so the sets {1,2,3} and {3,2,2,1} are equal.
If `x` is a member of set `S`, we write `x in S`. If `x` is not a member of set `S`, we write `x !in S`. There are a number of special sets with the following denotations:
Definition of subset
A subset of a set `B` is another set `A` such that:
`(AA x in A)(x in B)`
This definition says that all elements in `A` exist in `B`.
If `A` is a subset of `B`, we write `A sube B`. Note that a set is always a subset of itself, and `O/` is a
subset of every set. If `A` is a subset of `B` and `A != B`, then we write `A sub B` and say that `A` is a
proper subset of `B`. Notice how the `sub` and `sube` symbols resemble the `<` and `<=` symbols.
The negated forms of these symbols are `!sub` and `!sube`.
A superset of a set `B` is another set `A` such that `B sube A`. We can write `A supe B` to denote this. If `A` is a superset of `B` and `A != B`, we say `A` is a proper superset of `B`, and we write `A sup B`. The negated forms are `!sup` and `!supe`.
Definition of union
The union of two sets `A` and `B` is a set `C` such that:
`(AA x in C)(x in A or X in B)`
This definition says that the union of `A` and `B` contains all of the elements that exist in either `A` or `B`
(or both). The union of `A` and `B` is written `A uu B` (read "`A` union `B`"). Union is commutative,
associative, and distributive.
Definition of intersection
The intersection of two sets `A` and `B` is a set `C` such that:
`(AA x in C)(x in A and x in B)`
This definition says that the intersection of `A` and `B` contains all the elements that exist in both `A` and
`B`. The intersection of `A` and `B` is written `A nn B` (read "`A` intersect `B`"). Intersection is commutative,
associative, and distributive.
Definition of set difference
The difference of two sets `A` and `B` is a set `C` such that:
`(AA x in C)(x in A and x !in B)`
This definition says that the difference of `A` and `B` contains all elements that exist in `A` but not in `B`.
The difference of `A` and `B` is written `AB` (read "`A` minus `B`").
Definition of set complement
The complement of a set `A` is equal to `UUA`, where `UU` is the universe of discourse. The complement
of `A` is written `A'`. So if `x in A` then `x !in A'`, and vice versa.
Because the set complement operation depends on the universe of discourse, it is only
meaningful if a universe has been defined (or can be assumed).
Definition of power set
Given a set `A`, the power set of `A` (written `PS(A)`) is the set containing all subsets of `A`
(including `A` itself):
`PS(A) = {S:S sube A}`
Note that the power set of the empty set is a set containing the empty set (ie, {`O/`}), not the empty set
itself. Power set is distributive over intersection, but not over union.
Definition of cartesian product
The cartesian product of two sets `A` and `B` is a third set containing all possible ordered pairs
`(x,y)` where `x in A` and `y in B`.
`A xx B = {(x,y) : x in A and y in B}`
`A` and `B` are called factors of `A xx B`. The cartesian product is also known as the cross
product, or simply the product. The cartesian product is not associative or commutative, but
it is distributive.
More generally, the cartesian product of `N` sets `S_1 xx S_2 xx ... xx S_N` is a set containing all possible ordered lists `(c_1, c_2, ..., c_N)` where `c_i in S_i` (ie, `c_1 in S_1, c_2 in S_2`, etc).
Definition of cardinality
The cardinality of a set is the number of elements it contains. The cardinality of a set `A` is
written `A`.
Definition of disjoint
Two sets `A` and `B` are disjoint if they have no elements in comment. That is, if `A nn B = O/`.
Definition of interval
Given `S sube RR`, `S` is an interval if:
`(AA u,v in S)(AA x in RR)(u < x < v => x in S)`
By this definition, `O/` is not an interval, any set containing a single real number is an interval, and
any finite set containing more than one real number is not an interval (because given any two distinct
elements of the set, there would be a real number between them that is not in the set).
