APPLICATION OF SET

SET OF NUMBER

·         N= natural number -- {0, 1,2,3,4,...}

·         P = positive integer --{1,2,3,4,5,…}

·         Z = all integer – {…,-3,-2,-1 0,1,2,3,…}

·         R = all real number – {…,-1,-2.3,-1/3,0,1.2,3,3/4}

·         Null or Empty Set – denoted as { } or Ø

Set often denoted in notation = { :          }

Example :

A = { x : x ϵ P and 1 < x < 4}

Therefore : A = {2,3,}

INTERVAL

Interval is special subsets of R using some symbol.

Example a < b

·         (a,b) = {x ϵ R : a < x< b} -- open interval

·         [a,b] = {x ϵ R : a ≤ x ≤ b} – closed interval

·         [a,b) = {x ϵ R : a ≤ x < b}

·         (a,b] = {x ϵ R : a < x ≤ b}

POWER SET

·         The set of all subsets of a set S is = P(S) – power set of S

Rule : 

- P(S) denoted by |P(S)| , then if |S|=n, |P(S)|=2ⁿ

- if S = {a} then P(S)= {Ø , {a}}

Example :

S = {1,2,3}

Solution :

|S| = 2³ then |P(S)| = 8

P(S) = {Ø, {1},{2},{3},{1,2},{1,3},{2,3}{1,2,3}}

SPECIAL SET

∑ - is the whose element are symbols (include number) or letters, then

∑* - is the set of combination of all element from ∑.

RULE :

∑ = {a,b}

∑* = {a,b,ab,ba,aa,bb,aaa,aab,…}

To solve the problem, for every ∑* we introduce ‘ length ‘

For each w ϵ ∑*, length w is the number of elements in w.
length (ab) = 2 , (aab) = 3

EXAMPLE :

If ∑ = {a,b,c} , set A = { w ϵ ∑* : length (w) = 2}

Sol : A = {aa,ab,ba,dd,cc,ca,bc,cb,cc}