Chapter 2-sets.ppt - MetaLab by xiaoyounan

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									          CSNB 143
Discrete Mathematical Structures
           Chapter 2 - Sets
Sets
OBJECTIVES
 Student should be able to identify sets and its important
  components.
 Students should be able to apply set in daily lives.
 Students should know how to use set in its operations.
What, which, where, when
1. Basics of set      (Clear / Not Clear )
2.Terms used in set
 Equal sets          (Clear / Not Clear)
 Empty               (Clear / Not Clear)
 Disjoint / Joint    (Clear / Not Clear)
 Finite / Infinite   (Clear / Not Clear)
 Cardinality         (Clear / Not Clear)
 Subset              (Clear / Not Clear)
 Power set           (Clear / Not Clear)
3. Operations on sets
 Union                   (Clear / Not Clear)
 Intersection            (Clear / Not Clear)
 Complement              (Clear / Not Clear)
 Symmetric Difference    (Clear / Not Clear)

4. Venn Diagram
 Information Searching   (Clear / Not Clear)
SET
 A collection of data or objects.
 Each entity is called element or member, defined by symbol
  .
 Order is not important.
 Repeated element is not important.
 One way to describe set is to listing all the elements, in curly
  bracket.
 Ex 1:
A = {1, 2, 3, 4, 5}            B = {2, 3, 1, 4, 5)
C = {1, 2, 1, 3, 4, 5}
 Thus we said, sets A and B are equal. A = B
               1  A, 2  A, but 7  A
 How about A and C? How about B and C?
 Ex 2:
P = {p, q, r}            Q = {p, q, r, s}
R = {q, r, s}

So,    p  P,   q  P,          r  P,
       q  Q,   rR
Other way to describe set:
 A = {x| 1  x  5}
 A = {x| x is an integer from 1 to 5, both included}
 A = {x| x + 1 ; 0  x < 5}


 If the set has no element, it is called the empty set, denoted by
  {} or .

 Let   D = {6, 7, 8}
 So, A and D are called Disjoint Sets. Why? What is the example
  of joined set?
 A set A is called finite if it has n distinct elements, where n
   N (nonnegative number).
  Ex 3: A = {x| 1  x  5}

 The number of its elements, n is called the cardinality of A,
  denoted by |A|= 5.

 A set that is not finite is called infinite.
 Ex 4: A = {x| x ≥ 1}
Subset
 If every element of A is also an element of B, that is, if
  whatever x  A then x  B, we say that A is a subset of B
  or that A is contained in B, written as A  B (some books
  use symbol ).
 Sets that all its elements are part or overall of other set.
  Ex 5:
 A = {1, 2, 3, 4, 5}
 B = {1, 3, 5}
 C = {1, 2, 4, 6}, Thus,
B  A, but C  A
B  A but A  B
Consider Ex 1.
A = {1, 2, 3, 4, 5}      B = {2, 3, 1, 4, 5)
C = {1, 2, 1, 3, 4, 5}

 Is A  B?
 Is B  A?
 Is A  C?
 Is B  C?
Power Set
 If A is a set, then the set of all subsets of A is called the
  power set of A, denoted by P (A).
 A set that contains all its subset as its element.
  Ex 6:        A = {1, 2}
 P (A) = {{1}, {2}, {1, 2}, }
|P (A)| = 4
Operation on sets
Union
 Let say A and B are sets. Their union is a set consisting of all
  elements that belong to A OR B and denoted by A  B.
        A  B = {x|x  A or x  B}

Intersections
 Let say A and B are sets. Their intersection is a set
  consisting of all elements that belong to both A AND B and
  denoted by A  B.
        A  B = {x|x  A and x  B}
Operation on sets
Complement
 Let say set U is a universal set. U – A is called the
  complement of A, denoted by A’ (some book use A)
  A’ = { x|x  A}

 If A and B are two sets, the complement of B with
  respect to A is a set that contain all elements that belong to
  A but not to B, denoted by A – B.
  Try Ex 5. Find A – B, A – C, C – A, C – B.
Symmetric Difference
 Let say A and B are two sets. Their symmetric difference
  is a set that contain all elements that belong to A OR B but
  not to both A and B, denoted by A  B.
 A  B = {x|(x  A and x  B) or (x  B and x  A)}
  Try Ex 2. Find P  R.
Venn Diagram
Exercise :
 AB
 AB
 ABC
 ABC
 A–B
 B –A
 AB
Theorems
Commutative
 A B = B  A                      A  B = B A
Associative
 A  (B  C) = (A  B)  C
 A  (B  C) = (A  B)  C
Distributive
 A  (B  C) = (A  B)  (A  C)
 A  (B  C) = (A  B)  (A  C)
Idempotent
 A A =A                           A A =A
Complement
 A’’ = A              A  A’ = U
 A  A’ = 
 ’ = U               U’ = 
 (A  B)’ = A’  B’
 (A  B)’ = A’  B’
Universal Set
 AU=U                A  U =A
Empty Set
 A   =A             A=
2 disjoint sets
 |A  B| = |A| + |B|
2 joint sets
 |A  B| = |A| + |B| - |A  B|
3 disjoint sets
 |A  B  C| = |A| + |B| + |C|
3 joint sets
 |A  B  C| = |A| + |B| +|C| - |A  B| - |A  C| -
   |B  C| + |A  B  C|

								
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