MODULE 1

• Abstract Algebra


  study of algebraic structure
• Algebraic Structure


  consist of a set and binary operation
• <S,*>


  Where in S is the set, and * is the operation
• Algebraic Structures


  Associative Law holds all algebraic structures, but commutative law holds some algebraic structures
• Sets


  a well-defined collection of objects
• Null or Empty Set


  set with no elements
• Closure Property under multiplication


  If a, b ∈ Z, then a ∙ b ∈ Z
• Closure Property under addition


  If a, b ∈ Z, then a + b ∈ Z
• Commutative Property


  If a, b ∈ Z, then a +/x b = b +/x a
• Associative Property


  If a, b, c ∈ Z, then a +/x (b +/x c) = (a +/x b) +/x c
• Existence of additive identity


  a + 0 = a
• Existence of Multiplicative identity


  a ∙ 1 = a
• Existence of additive inverse


  a + x = 0 , namely (x = −a)
• Cancellation Law


  a ∙ c = b ∙ c implies a = b
• Distributive Law


  a (b + c) = a ∙ b + a ∙ c
• Trichotomy Law


  For any two integers a, b ∈ Z, exactly one is true
• Well Ordering Principle


  Every nonempty set of positive integers contains a smallest member.
• 2 steps of Mathematical Induction

  1. Basis Step
  2. Inductive Step
• Division Algorithm


  If a and b are any positive integers, then there exist unique integers q and r, with 0 ≤ r < b, such that a = qb + r.
• Division Algorithm


  a = qb + r

  a = largest integer

  q = quotient

  b = smaller integer

  r = remainder
• Linear Combination


  d = ma + nb
• Subsets


  are the cells of the partition.
• Properties of Equivalence Relation

  1. Reflexive Property
  2. Symmetric Property
  3. Transitive Property
• Function or Mapping


  a rule that assign to each element of X exactly one element of Y.
• Properties of Binary Operation

  1. Commutative Operation
  2. Associative Operation