Chapter 5 Abstract Algebra - BUM1223 APPLICATIONS...

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BUM1223 D ISCRETE M ATHEMATICS & A PPLICATIONS C HAPTER 5: A BSTRACT A LGEBRA Dr. Mohd Sham Mohamad ([email protected] ) Chapter 3: Graphs
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5.1 Groups 5.2 Abelian Group 5.3 Semigroups and Monoid 5.4 Subgroups 5.5 Cyclic Groups 5.6 Rings 5.7 Commutative Rings 5.8 Field Chapter 3: Graphs CHAPTER 5: ABSTRACT ALGEBRA
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Lesson Outcome: Define properties of a group Chapter 3: Graphs 5.1 GROUPS
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Chapter 3: Graphs A binary operation on a non-empty set A is a map : f A A A such that (i) f is defined for every pair of elements in A (ii) f uniquely associates each pair of elements in A to some element of A i.e. : , f a b a b c A , , a b A Definition (Binary relation)
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Definition (Groups) Chapter 3: Graphs Let G be a nonempty set with a binary operation . G is a group if closed under operation and satisfies the following properties: (i) Associativity a b c a b c , , , a b c G (ii) Identity There exist a unique identity e G such that a e e a a , a G   (iii) Inverse a G   there exist a unique b G such that a b b a e (denote as 1 b a )
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Example 5.1 Chapter 3: Graphs . The set of integers , the set of rational numbers and the set of real numbers are all groups under ordinary addition. Closed Operation: Addition Closed Associative a b c a b c , , , a b c G by properties of addition in , and . Identity There exist a unique identity 0 G such that 0 0 a a a , a G   . Inverse a G   there exist a unique b G such that 0 a a a a    
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Example 5.2 Chapter 3: Graphs The set of positive rational numbers under multiplication is a group. Closed Operation: Multiplication Closed Associative ab c a bc , , , a b c G by properties of multiplication . Identity There exist a unique identity 1 G such that 1 1 a a a , a G   . Inverse a G   there exist a unique 1 b G a such that 1 1 1 a a a a
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Example 5.3 Chapter 3: Graphs The subset 1, 1, , i i of the complex numbers is a group under complex multiplication. Closed Operation: Multiplication Closed (Multiplication Table) 1 -1 i i 1 1 -1 i i -1 -1 1 i i i i i -1 1 i i i 1 -1 Associative ab c a bc , , , a b c G since commutative ab ba . Identity There exist a unique identity 1 G such that 1 1 a a a , a G   . Inverse 1 1 1 1 1 1,( 1) 1, , i i i i   .
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Exercise 5.1 Chapter 3: Graphs 1. Show that the set 0,1,2,..., 1 n n for 1 n is a group under addition modulo n .
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