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Lec07_Algebraic_Structures

# Lec07_Algebraic_Structures - TDS1191 Discrete Structures...

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[Lecture07][Algebraic Structures] Discrete Structures 1 TDS1191 Discrete Structures Lecture 07 Algebraic Structures Multimedia University Trimester 2, Session 2011/2012

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[Lecture07][Algebraic Structures] Discrete Structures 2 These are the keywords that you will learn in this lecture… Algebraic Structures Semigroups Monoids Groups Abelian Groups Rings Fields
[Lecture07][Algebraic Structures] Discrete Structures 3 Algebraic Structures are used in: Cryptography Finite-State Machines Vector Spaces Linear Transformations Solving Systems of Equations

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[Lecture07][Algebraic Structures] Discrete Structures 4 Introduction to Algebraic Structures Algebraic Structures Any formal mathematical systems consisting of a set of objects and operations on those objects. Examples: Structures Objects Operations Propositions , , ¬ , , , Set Algebra Sets , , - Numerical Algebra Real numbers +, -, × , ÷ Boolean Algebra Digits 0 & 1 +, ·, - This one has no formal name. Hence, the space is left blank intentionally.
[Lecture07][Algebraic Structures] Discrete Structures 5 Operations on Structures Let S be a nonempty set. An n -ary operation is a mapping between set S of n order to another set S , denoted as f: S × S × S × S × … S S Let S be a nonempty set. An n -ary operation is a mapping between set S of n order to another set S , denoted as f: S × S × S × S × … S S Hence, for n = 1, f: S S is called a unary operation. for n = 2, f: S 2 S is called a binary operation. A non-empty set together with one or several operations is an algebraic structure. n times Or can be compactly written as, f: S n S

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[Lecture07][Algebraic Structures] Discrete Structures 6 General Notation of An Algebraic Structure An algebraic structure with set S and operation * is denoted as ( S ,*) or just S . For a binary operation * of set S, if for all a , b S , a * b S . An algebraic structure with set S and operation * is denoted as ( S ,*) or just S . For a binary operation * of set S, if for all a , b S , a * b S . Note that * is any operation on set S , it is NOT a multiplication operation. This is read as: a “operation” b
[Lecture07][Algebraic Structures] Discrete Structures 7 A binary operation (*) can be defined by means of a table if the set is finite. Example: A set S = { a , b , c , d } with binary operation * The entry in the row labeled a and the column labeled b is a * b a * b = c An Example of a Tabled Binary Operation * a b c d a a c d b b c d a b c d a b c d b b c a S

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[Lecture07][Algebraic Structures] Discrete Structures 8 Properties of Structures An algebraic structure may have these properties: - closure - commutative - associative - distributive - identity - inverses - idempotent - An algebraic structure may have these properties: - closure - commutative - associative - distributive - identity - inverses - idempotent - Closure Property If an operation on the members of a set produces images which are also members of the same set, then the set is said to be closed under that operation, and this property is called the closure property.
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Lec07_Algebraic_Structures - TDS1191 Discrete Structures...

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