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					SETS                                    NOTE : Set A can be substituted with ANY   Commutative Laws
Symbols and notations                   other set or combination of sets           a) A B                 b)   A B 
:            :                        Of apples and oranges                      *
*             *                          A   A
                                        * “  ”        ,   A                      Associative Laws
:                 :                                                              a)  A  B   C 
*                  *

:                 :                   B                                       b)  A  B   C 
*                  *
                                        A  B                                 *
                                                                                   *
                                        A   
                                         A  B  C    
A :                                                                               Bracket Matters
                                          
*                                                                                            
                                                                                   *Order -  ...  
                                                                                   *Only 2 sets in a bracket, every pair of sets
SET ALGEBRA                             A  A                                   MUST have bracket
Identity & Inverse Laws

a) A  A 
                                                                                   A  B  A  B
                        b)   A A      B  B 
*with itself
                                         A  B   A  B                       A  B  C   A  B  C  
                                                                                                                    
                                                                                                                    
c) A                 d) A                                                    A B C
*with nothing                                                                      A  B  B
                                        Exercise 1
e) A                 f) A         1) B  C                               A  B   B  C   C  D
*with everything                        2)     
g) A  A              h)   A  A    3)  A  B   A  B                   Combining Commutative and Associative
                                        4)  A  B    A  B  
*inside with outside                                                               *Can change position AND order if SAME
                                        5)  A  B   
                                                                                   operation
i)  A''                                                                         *NEVER do that when DIFFERENT
*outside of outside                     6)  A  B    A  B                   operation or when have complement
                                                                                   outside bracket
                                             B   
                                                                                   A   A  B 
                                        7)
j)                  k)   
*complement of nothing                  8)  A  B   A  B                    A  B  B  A
*complement of everything                                                         A  B  C 
                                        9) B 
                                                                                                
                                                                                   A  B  C 

                                                                                                                                   1
Example                                   a)                                          Associative vs Distributive vs Identity
A  B  A                                                                          A  B  C 
                                          b)
                                                                                      *
                                          Identify the form which can be „factored‟   A  B  C 
                                           A  B   A  C                         *

 A  B     B  A                     A  B   B  A                       A   A  B  
                                           A  B    A  B                       A   A  B  
                                          Simplify
                                          B  A  C  A                         A  B    A  B 
                                                                                       A  B    A  B 
                                           A  B   B  C                        A  B  A  B
Exercise 2 (Simplify using Set Algebra)
                                                                                       A  B    A  B 
1) A  B   A
2)  A  B   B                         A   B  A 
                                                                                       A  B    A  B 
3)  A  B   A  B 
4)  A  B   B  C   C  A

Distributive Laws                          A  B   B   A
a) A  B  C  
b) A  B  C  
*

                                          Exercise 3 (Simplify using Set Algebra)
How to expand
*Be VERY careful of operation
                                          1) A   A  B                             A  B   B  A'
A  B  C    A  B    A  C       2)  A  B   B

A  B  C    A  B    A  C       3) B    A  B 
                                          4)  A  B   A
*Rearrange using commutative law          5)  A  B    A  B 
BEFORE expanding
                                          6)  A  B    A  B 
B  C   A  B  A  C  A          7) B  C   C 'B
B  C   A                             8) C  A   A  C 

How to factor
                                                                                                                                2
                                                                                            A  B   A  B
Exercise 4 (Simplify using Set Algebra)   De Morgan’s Law
1)  A  B    A  B                               
                                          a) A  B 
2)  A  B    A  B 
                                                      
                                          b)  A  B                                      A  B    A  B  
3)  A  B    A  B 
4) B  C   B   A
                                          *
                                          NOT
5) C  B   B   A
6)  A  C    A  B                                                                   Exercise 6 (Simplify using Set Algebra)
                                          *REMEMBER to CHANGE operation                                          
                                                                                                A   A  B 
                                                                                        1)
Exercise 5 (Simplify using Set Algebra)   A  B                          A  B 
(DO without referring to any notes)                                                             B   A  A
                                           A  B                       A  B 
                                                                                           2)
1)  A  B    A  B                                                                                       
                                                                                                A   A  B
                                           A  B                      A  B 
                                                                                           3)
2)   A  B   A  B                                                                   4)    A  B    A  B  
3) C  A   A  C 
                                                                                          5)   B  A  B  A
4)  A  B    A  B 
5) A  B   A
                                          Order of brackets Important                      6)    A  B   A  B
                                                               
6) A   A  B                          A  B  C                                   7)   A  A  B   C 
                                                                                                                       
                                                                   
7)  A  B    A  B                                                                   8)    A  B  C   A
8)  A  B    A  B                                            
                                          A  B  C              
9)  A  B   B                                                      
                                                                                           Adsorption Law (maybe not in syllabus,
10)  A  B    A  B 
                                                                                           use only if absolutely necessary)
                                          Don‟t mix with other laws, must break it first   a) A   A  B  
11) B  C                             BEFORE applying other laws                       b) A   A  B  
12) B  C   B   A                                   
                                          A   A  B   A  A  B                    *
13)  A  B    A  B                                                                  *
14)  A  B   B
                                           A  B   A  B  A  B  B
                                                                                           Adsorption Law CAN‟T be proved from the
15)  A  B   A  B                                                                  other laws, thus it is very important to
16) C  B   B   A                                                                  identify it
                                          A  B  A                                    A  A  B 
17)  A  B   A
18)  A  C    A  B 




                                                                                                                                      3
A  A  B                               Difference of Sets                    Exercise 8 (Simplify using Set Algebra)
                                          A B                                 1)  A  B   A         2) B   A  B 
A  A  C                               *this isn‟t a law, but an operation                                                  
A  A  B  C                        *always apply this BEFORE any law     3)  A  B    A  B  4)     A   A  B
A   A  B                                                                  5) A   A  B               6)  A  B   B
A  B  B                                                                     7)  A  B    A  B 
                                          Of apples and oranges

 A  B    A  B   C                                                     8)  A  B   C  B  9)  A  B    A  C 
                                           B A                                Proving identities (General guide)
 A  B   A  B                       A  B                               1) Use correct structure of prove
                                                                                2) Don‟t skip steps
                                          A  ( B  C ) 
                                                                                3) NEVER use own laws
                                          A  ( B  C )                      4) Keep an eye on the other side
                                          A  B  A  B                    5) Start from more complicated side, when
                                                                                   stuck, try from the other side
                                          A  B  C 
Compare
A  A  B                                                                     6) If something seems wrong, stop and

A   A  B 
                                                                                   check above workings
                                          A  B  C  
A   A  B                                                                   Using set algebra, prove   B   B   A  A  B
A  A  B                                A  B    B  A 
 A  B    A'B                                                             Exercise 9 (Prove using Set Algebra)
                                                                                1)  A  B    A  B   
 A'B    A'B 
                                                                                2) A  B  A  
* Do note use adsorption law unless
absolutely necessary as it might not be
                                           A  B    A  B               3) A  B  B  A
accepted in STPM marking scheme.                                                4)  A  B    A  B   A  B

Exercise 7 (Simplify using Set Algebra)                                         5)  A  B  B  A  A
1)  A  B   B                                                                                             
                                                                                6)  A  B   A   A  B 
2) A  B   A
                                                                                                   
3) A  B  A                                                                7) A   A  B   B  

                  
4) A  B  C   A                                                                                    
                                                                                8) B   A  B   A  
5)  A  B    A  B   C                                                  9) A  B  C    A  B    A  C 
6)  A  B    A  B                                                         10) A  B  C    A  B    A  C 
7)  A  B    A  B                                                       11) A  B  C    A  B    A  C 
8)  A  B   B  C                                                          12)  A  B   C   A  C   B  C 


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posted:8/15/2011
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