Math tips, Math formulas

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 .


 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

Operations

1. 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, .
2. The intersection of two sets is the set containing all elements of both and . This is written as . For example, .
3. 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.