# Set

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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 

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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 

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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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