## org.metasyntactic.math.algebra Interface Set

All Superinterfaces:
java.util.Collection, java.util.Set
All Known Subinterfaces:
AbelianGroup, Field, Group, Groupoid, Monoid, Ring, Semigroup
All Known Implementing Classes:
AbstractField

public interface Set
extends java.util.Set

A set is a collection of distinguishable objects, called its members or elements. If an object x is a member of a set S, we write x ∈ S (read "x is a member of S" or, more briefly, "x is in S"). If x is not a member of S, we write x ∉ S. We can describe a set by explicitly listing its members as a list inside braces. For example, we can define a set S to contain precisely the numbers 1, 2, and 3 by writing S = {1, 2, 3}. Since 2 is a member of the set S we can write 2 ∈ S, and since 4 is not a member, we have 4 ∉ S. A set cannot contain the same object more then once, and its elements are not ordered. Two sets A and B are equal, written A = B, if they contain the same elements. For example, {1, 2, 3, 1} = {1, 2, 3} = {3, 2, 1}.

We adopt special notations for frequently encountered sets.

• ∅ denotes the empty set, that is, the set containing no members.

For any set A, we have AA. For two sets A and B, we have A = B if and only if AB and BA. For any three sets A, B, and C, if AB and BC, then AC. For any set A we have ∅ ⊆ A.

 Method Summary ` Set` `difference(Set c)`            The difference of sets A and B is the set ` Set` `intersection(Set c)`            The intersection of sets A and B is the set ` Set` `union(Set c)`            The union of sets A and B is the set

 Methods inherited from interface java.util.Set `add, addAll, clear, contains, containsAll, equals, hashCode, isEmpty, iterator, remove, removeAll, retainAll, size, toArray, toArray`

 Method Detail

### intersection

`public Set intersection(Set c)`

The intersection of sets A and B is the set

AB = {x | x ∈ A and x ∈ B}.

Note! Neither set is affected by this operation. To make this set equal to the intersection of itself and c use retainAll(Collection c)

Returns:
The set equal to the intersection of this set and c
`Set.retainAll(Collection c)`

### union

`public Set union(Set c)`

The union of sets A and B is the set

AB = {x | x ∈ A or x ∈ B}.

Note! Neither set is affected by this operation. TO make this set equal to the union of itself and c use addAll(Collection c)

Returns:
the set equal to the union of this and c
`Set.addAll(Collection c)`

### difference

`public Set difference(Set c)`

The difference of sets A and B is the set

A - B = {x | x ∈ A and x ∉ B}.

Note! Neither set is affected by this operation. To make this set equal to the difference of itself and c use removeAll(Collection c)

Returns:
the Set equal to the difference of this and c
`Set.removeAll(Collection c)`