Senin, 14 November 2011

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

SET

If A denotes as a set and if x is an element, we shall write.
x∈A
as an abbreviation for the statement that x is an element of A, or that x is a member of A, or that x belongs to A, or that the set A contains the element x, or that x is in A. If x is an element that does not belong to A, we shall write.
x∉A
If A and B are sets such that x∈A implies that x∈B (that is every element of A is also an element of B), then we shall say that A is contained in B, or that B contains A, or that A is a subset of B, and we shall write.
A⊆B or B⊇A
Definition, two sets A and B are equal if they contain the same element. If the sets A and B are equal, we write A=B

Examples:
(a) The set {x∈N : x2‒3x+2=0}
consists of those natural numbers satisfying the stated equation since the only solutions of the quadratic equation x2‒3x+2=0 are x=1 and x=2, instead of writing the above expression we ordinarily denoted this set by {1,2}, thereby l;isting the elements of the set.
(b) Sometimes of formula can be used to abbreviate the description of a set. For example, the set of all even natural numbers could be denoted by {2x : x∈N}, instead of the more cumbersome {y∈N : y=2x, x∈N}
(c) The set {x∈N : 6<x<9} can be written explicitly as {7,8}, thereby exhibiting the elements of the set. Of course, there are many other possible description of this set, for example:
{x∈N : 40<x2<80}
{x∈N : x2‒15x+56=0}
{7+x : x=0 or x=1}