MAE 121 Order Properties of the System (ℤ, +) Lecture 2 PDF

SCHOOL OF SCIENCE AND MATHEMATICS EDUCATION Email: [email protected] MAE 121 Order Properties of the System (ℤ, +) Exploring Integer Properties through Formal Proofs Mr. W Mangcengeza Lecture 2 ...

SCHOOL OF SCIENCE AND MATHEMATICS EDUCATION Email: [email protected] MAE 121 Order Properties of the System (ℤ, +) Exploring Integer Properties through Formal Proofs Mr. W Mangcengeza Lecture 2  Introduction Understanding order properties is fundamental in mathematics as they support key operations and concepts. These properties, such as commutative, associative, and distributive, are essential for simplifying expressions, solving equations, and ensuring consistency in calculations. They also underpin more advanced mathematical structures and enhance problem-solving and logical reasoning skills. Mastery of these properties is crucial for both basic arithmetic and advanced mathematics.  Trichotomy Property Statement: For any a, b ∈ ℤ, one and only one of the following is true: a < b, a = b, or a > b. Example: a = -3, b = 2 Proof outline Proof of Trichotomy Property For a, b ∈ ℤ, consider d = a - b d > 0 implies a > b d = 0 implies a = b d < 0 implies a < b One and only one of these cases is true  Transitivity Property Statement: For any a, b, c ∈ ℤ, if a < b and b < c, then a < c. Example: a = -5, b = 0, c = 4 Proof outline Proof of Transitivity Property Assume a < b and b < c b = a + m and c = b + n c=a+m+n Since m and n are positive, a < c Proof of Transitivity Property  Antisymmetry Property Statement: For any a, b ∈ ℤ, if a < b, then it is not the case that b < a. If a ≤ b and b ≤ a, then a = b. Example: a = 3, b = 7 Proof outline  Proof of Antisymmetry Property Assume a < b There exists m > 0 such that b = a + m Assume b < a There exists n > 0 such that a = b + n a = a + m + n implies 0 = m + n Contradiction, thus a < b implies ¬(b < a)  Addition Compatibility Property Statement: For any a, b, c ∈ ℤ, if a < b, then a + c < b + c. Example: a = -2, b = 1, c = 3 Proof outline  Proof of Addition Compatibility Property Assume a < b There exists m > 0 such that b = a + m Add c to both sides: b + c = a + c + m Since m > 0, a + c < b + c  Multiplication Compatibility Property Statement: For any a, b, c ∈ ℤ, if a < b and c > 0, then a · c < b · c. Example: a = 1, b = 4, c = 2 Proof outline Proof of Multiplication Compatibility Property Assume a < b and c > 0 There exists m > 0 such that b = a + m Multiply by c: b · c = a · c + m · c Since m, c > 0, a · c < b · c  Reflection Questions Why is the trichotomy property crucial for defining a total order on ℤ? How does the transitivity property help in reasoning about the relative order of multiple integers? Can you think of a situation where the antisymmetry property helps in proving equality? Why is it important for the addition compatibility property to hold in ordered sets? What might happen if the multiplication compatibility property did not hold for positive integers?