Set
From Mathematics
A set is a collection of distinct, unordered objects. Sets are typically collections of numbers, though a set may contain any type of data (including other sets).
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[edit] Set notation
Sets are notated using french braces (
) with elements delimited by commas. There are three ways to represent a set.
- Strict enumeration - each element in a set is explicitly stated (e.g.,
).
- Pattern enumeration - sets with elements following a clear pattern can be shortened from strict enumeration by only showing enough elements to describe the pattern and representing the rest with an ellipsis (e.g.,
).
- Set former (or set builder)- elements in a set are defined as a function of one or more variables in a given domain that meets a condition. The presence of a condition is optional. Some syntaxes and variations for a set former are as follows:
For example, 
defines the set of integers 1 through 10.
, given a function 
and predicate 
, the set of all values 
for which 
is true.
, given a set 
and predicate , a subset of all 
in for which is true.
[edit] Set properties and operations
Several properties and operations have been defined for sets. For the purpose of this section, sets are assumed to be collections of numbers. Set is defined as the set 
.
[edit] Properties
- An object is an element of a set when it is contained in the set. For example, 1 is an element of . This is written as
. Similarly, the fact that 11 is not an element of is written as 
. The universe (usually represented as 
) is a set containing all possible elements, while the empty set or null set (represented as 
or 
) is a set containing no elements.
- The cardinality of a set is the number of elements in the set. The cardinality of (written as
or 
) is 10.
- The complement of a set is the set containing all elements of the universe which are not elements of the original set. For example, if the universe is defined as
, then the complement of with respect to (written as 
) is 
. The cardinalities of a set and its complement together equal the cardinality of the universe. Thus, the universe and the null set are complements of each other.
- A set is a subset of another when all the elements in the first set are contained in the second set. Given sets
and 
, is a subset of , notated as 
, if and only if for all , 
implies 
. All sets are subsets of the universe. By definition, all sets are subsets of themselves and by convention, the null set is a subset of all sets. For example, 
. Any given set has 
subsets.
- Two sets are equal if they are subsets of each other.
- A set's proper subsets are all subsets except the set itself. This relationship is notated by
[edit] Operations
- The union of two sets is the set containing all elements of either or , including elements of both and . This operation is written as
. For example, 
.
- The intersection of two sets is the set containing all elements of both and . This is written as
. For example, 
. The sum of the cardinalities of the intersection and union of two sets is equal to the sum of the cardinalities of the two sets.
[edit] Other functions on sets
Some functions on sets return a set which may not necessarily be a subset of the universal set. Given sets 
and :
- The Power set of , denoted
, is the set containing all subsets of .
- The Cartesian product of and , denoted
, is the set of ordered pairs 
where 
and 
. That is, 
.
