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

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

Description

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.

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