Operation of Sets

1.1 Complement

The relative complement set of set  A to set  B consists of the elements which are in B but not in A. The complement set can be defined by the

following formula.

If the set B is the universal set X, then this kind of complement is an absolute complement set Ᾱ. That is,  Ᾱ  = X – A

In general, a complement set means the absolute complement set. The  complement set is always involutive 

The complement of an empty set is the universal set, and vice versa.

1.2 Union

The union of sets A and B is defined by the collection of whole elements of A and B.

The union might be defined among multiple sets. For example, the union of the sets in the following family can be defined as follows.

where the family of sets is {Ai | i ∊ I}.

The union of certain set A and universal set X is reduced to the universal set.

                         A∪ X = X

The union of certain set A and empty set ⏀‡is A.

                                                                    A ∪⏀= A

The union of set A and its complement set is the universal set

                                                                      A ∪ Ᾱ = X.

1.3 Intersection

The intersection A ⋂ B consists of whose elements are commonly included  in both sets A and B.

The Intersection can be generalized between the sets in a family of sets

The intersection between set A and universal set X is A.

                        A ⋂ X = A.

The intersection of A and empty set is empty set

                                          A⋂ ⏀=⏀ ‡.

The intersection of A and its complement is all the time empty set

                                              A ⋂ Ᾱ = ⏀.

When two sets A and B have nothing in common, the relation is called as disjoint. Namely, it is when the intersection of A and B is empty set

                                               A ⋂ˆ B = ⏀‡.

1.4 Partition of Set

Definition (Partition)  A decomposition of set A into disjoint subsets whose union builds the set A is referred to a partition. Suppose a partition

of A is π ,

then A_i satisfies following three conditions.

If there is no condition of (2), π (A) becomes a cover or covering of the set A.

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Kwang H. Lee, First Course on Fuzzy Theory and Applications, 2005, Springer, pag(3-5)