Order Properties of the System (ℤ, +)

Order Properties of the System (ℤ, +)

• Order properties are important for understanding mathematical operations and concepts.
• They are essential for simplifying expressions, solving equations, and ensuring consistency in calculations.

Trichotomy Property

• Statement: For any two integers, a and b, one and only one of the following statements must be true: a is less than b, a is equal to b, or a is greater than b.
• Proof:
  • Consider the difference d = a - b.
  • If d is greater than zero, then a is greater than b.
  • If d is equal to zero, then a is equal to b.
  • If d is less than zero, then a is less than b.
  • Only one of these cases can be true because the difference d can only have one value.

Transitivity Property

• Statement: If a is less than b and b is less than c, then a is less than c.
• Proof:
  • Assume a is less than b and b is less than c.
  • This means there exist positive integers m and n such that b = a + m and c = b + n.
  • Substituting b in the second equation, we get c = a + m + n.
  • Since m and n are positive, a must be less than c.

Antisymmetry Property

• Statement: If a is less than b, then it is not the case that b is less than a.
• Proof:
  • Assume a is less than b.
  • This means there exists a positive integer m such that b = a + m.
  • Assume b is less than a.
  • This means there exists a positive integer n such that a = b + n.
  • Substituting b from the first equation into the second equation, we get a = a + m + n, which implies 0 = m + n.
  • This is a contradiction because m and n are positive integers.
  • Therefore, if a is less than b, then b cannot be less than a.

Addition Compatibility Property

• Statement: If a is less than b, then a + c is less than b + c for any integer c.
• Proof:
  • Assume a is less than b, so there exists a positive integer m such that b = a + m.
  • Adding c to both sides, we get b + c = a + c + m.
  • Since m is positive, a + c is less than b + c.

Multiplication Compatibility Property

• Statement: If a is less than b and c is greater than zero, then a · c is less than b · c.

Explore the crucial order properties of integers with this quiz. Learn about the trichotomy and transitivity properties, which are fundamental for mathematical reasoning. Test your understanding of how these properties apply to simplifying expressions and solving equations.