SET OPERATIONS

We know define basic methods of constructing new sets from given ones.
Definition
(a) If A and B are sets, then their intersection, denote by A∩B, is the set of all elements that belong to both A and B. In other words, we have

A∩B={x : x∈A and x∈B}

(b) The union of A and B, denoted by A∪B, is the set of all elements that belong to either A or B. In other words, we have

A∪B={x : x∈A or x∈B}

In connection with the union of two sets, it is important to be aware of the fact that the word "or" is being used in the inclusive sense. To say x belongs to A or B allows the possibility that x belongs to both sets. In legal terminology this inclusive sense is sometimes indicated by "and/or"
Definition
the set that has no elements is called the empty or the void set and will be denoted by the symbol ∅. If A and B one sets with no elements (that is, if A∩B=∅), then we say that A and B are disjoint or that thay are nonintersecting.
Theorem
Let A, B, C be any sets then
(a) A∩A=A, A∪A=A,
(b) A∩B=B∩A, A∪B=B∪A,
(c) (A∩B)∩C=A∩(B∩C),
      (A∪B)∪C=A∪(B∪C),
(d) A∩(B∪C)=(A∩B)∪(A∩C),
      A∪(B∩C)=(A∪B)∩(A∪C)


Buktikan A∩A=A
Untuk membuktikan A∩A=A diperlukan langkah sbb.
(i) Ditunjukkan A∩A⊆A
(ii) Ditunjukkan A⊆A∩A
Penyelesaian:
(i) Ditunjukkan A∩A⊆A

     Ambil sembarang x∈A∩A akan ditunjukkan x∈A

     Jelas  x∈A∩A