Order Properties of the System (ℤ, +)

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Questions and Answers

The Trichotomy Property states that for any integers a and b, either a < b, a = b, or a > b is true.

True (A)

The Transitivity Property states that if a < b and b < c, then b < a.

False (B)

The Addition Compatibility Property indicates that if a < b, then a + c < b + c holds true for any integer c.

True (A)

The Antisymmetry Property states that for any integers a and b, if a < b, then it must be the case that b < a.

<p>False (B)</p> Signup and view all the answers

If a = -3 and b = 2, the Transitivity Property can be used to assert relations with another integer c.

<p>True (A)</p> Signup and view all the answers

According to the Antisymmetry Property, if a ≤ b and b ≤ a, then a must be less than or equal to b.

<p>False (B)</p> Signup and view all the answers

If c > 0, then for a < b, it follows that a · c < b · c according to the Multiplication Compatibility Property.

<p>True (A)</p> Signup and view all the answers

The proof of the Addition Compatibility Property shows that adding the same number to both sides of an inequality will reverse its order.

<p>False (B)</p> Signup and view all the answers

For the integers a = -5, b = 0, and c = 4, the Transitivity Property can demonstrate that -5 < 4.

<p>True (A)</p> Signup and view all the answers

Under the Addition Compatibility Property, for any integers a and b, a + c < b + c requires a < b.

<p>True (A)</p> Signup and view all the answers

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Study Notes

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.

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

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