SET OPERATIONS

We now define basic methods of constructing new sets from given ones.
112. Definition

1. If A and B are sets, then their intersection denote by A∩B, is the set of all element that belong to both A and B. In other words, we have 

   A∩B = {x: x∈A and x∈B}

2. The union of A and B denoted by A∪B, is the set of all element that belong to either A or B. In oher words we have  

   A∪B = {x: x∈A or x∈B}

In connection with the union of two sets, if is important to be aware of the fact that the words "or" is being used in the inclusive sense. To say x belong 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".


113. Definition The set that no elements is called the empty or the void set and will be denoted by the symbol ∅. If A and B are sets with no common elements (that is, if A∩B = ∅) then we say that A and B are disjoint or that they arenon intersecting.
114 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