August 21, 2017 05:03:11
Math
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Math > Sets > Definitions

Definitions

The definitions of sets and related terms.

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Definition of a set [tags: set emptySet U]
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:
  • `O/`: This represents the empty set (also known as the null set). The empty set has no elements.
  • `NN`: This represents the natural numbers (also known as counting numbers). Unfortunately, there is no general agreement as to this set's exact definition. It can refer to either the set of positive integers or the set of non-negative integers. In contexts where `NN` represents the non-negative integers, `PP` often represents the positive integers (however, a few authors even use `PP` to represent the non-negative integers). Where `NN` represents the positive integers, `WW` often represents the non-negative integers. `NN_0` is sometimes used to indicate the non-negative integers, and `NN^**` or `NN^+` are sometimes used to indicate the positive integers.

    To avoid ambiguity, the following symbols are recommended:
    SetSymbol
    integers`ZZ`
    positive integers`ZZ^+`
    non-negative integers`ZZ^**`
    negative integers`ZZ^-`
    On this web site, `NN` will denote the positive integers.
  • `WW`: This represents the whole numbers, which are usually defined to be the non-negative integers. But `WW` is occasionally used to represent the positive integers, so to avoid ambiguity it may be best to use the symbols above. On this web site, `WW` represents the non-negative integers.
  • `QQ`: This represents the rational numbers, which are all the numbers that can be represented as an integer divided by a natural number (a positive integer).
  • `RR`: This represents the real numbers.
  • `CC`: This represents the complex numbers.
  • `UU`: This is the universe of discourse, which is a set that represents the overall framework of the given situation. Usually this is some standard set such as `RR` or `ZZ`.
Notation such as `QQ^+` for positive rational numbers or `RR^(>=0)` for non-negative real numbers is sometimes used.
Definition of subset [tags: subset properSubset superset properSuperset]
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 [tags: 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 [tags: 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 `A-B` (read "`A` minus `B`").
Definition of set complement [tags: U]
The complement of a set `A` is equal to `UU-A`, 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 [tags: powerSet]
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 [tags: setProduct setFactor]
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 [tags: 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 [tags: 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 [tags: 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).
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