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Questions and Answers
What is the statement of the Trichotomy Property?
What is the statement of the Trichotomy Property?
- For any a, b ∈ ℤ, both a and b must be integers.
- For any a, b ∈ ℤ, a ≤ b, b ≤ a, and a ≠b.
- For any a, b, c ∈ ℤ, if a < c, then b < c.
- For any a, b ∈ ℤ, one and only one of the following is true: a < b, a = b, or a > b. (correct)
Which of the following correctly describes the Transitivity Property?
Which of the following correctly describes the Transitivity Property?
- If a < b and b < c, then a < c. (correct)
- If a > b, then it implies b < a.
- If a ≤ b and b ≤ c, then a = c.
- If a < c, then it must be that c < a.
What does the Antisymmetry Property imply?
What does the Antisymmetry Property imply?
- If a < b, then b < a must also be true.
- If a and b are equal, then a < b must also hold.
- If a ≤ b and b ≤ a, then a must be equal to b. (correct)
- If a > b, then a must be less than b.
What conclusion can be drawn from the Proof of Addition Compatibility?
What conclusion can be drawn from the Proof of Addition Compatibility?
Which property asserts that multiplying both sides of an inequality by a positive integer maintains the order?
Which property asserts that multiplying both sides of an inequality by a positive integer maintains the order?
If d = a - b and d < 0, what can be concluded about a and b?
If d = a - b and d < 0, what can be concluded about a and b?
In the proof of the Antisymmetry Property, what leads to a contradiction?
In the proof of the Antisymmetry Property, what leads to a contradiction?
Which of the following examples illustrates the Addition Compatibility Property?
Which of the following examples illustrates the Addition Compatibility Property?
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Study Notes
Order Properties of the System (ℤ, +)
- The order properties of the system of integers are essential for mathematical understanding and operations.
- They allow for simplifying expressions, solving equations, and ensuring consistency in calculations.
- Mastering these properties is crucial for both basic arithmetic and advanced mathematics.
Trichotomy Property
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For any two integers, a and b, one and only one of the following relationships holds:
- a < b (a is less than b)
- a = b (a is equal to b)
- a > b (a is greater than b)
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Proof:
- Consider the difference d = a - b
- If d > 0, then a > b
- If d = 0, then a = b
- If d < 0, then a < b
Transitivity Property
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For any three integers, a, b, and c, if a < b and b < c, then a < c.
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Proof:
- Assume a < b and b < c.
- There exist positive integers m and n such that b = a + m and c = b + n.
- Substituting, we get c = a + m + n.
- Since m and n are positive, a < c.
Antisymmetry Property
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For any two integers, a and b, if a < b, then it is not the case that b < a.
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If a ≤ b and b ≤ a, then a = b.
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Proof:
- Assume a < b.
- There exists a positive integer m such that b = a + m.
- Assume b < a.
- There exists a positive integer n such that a = b + n.
- Substituting, we get a = a + m + n, which implies 0 = m + n.
- This is a contradiction, as m and n are positive integers.
- Therefore, a < b implies ¬(b < a).
Addition Compatibility Property
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For any three integers a, b, and c, if a < b, then a + c < b + c.
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Proof:
- Assume a < b.
- 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 > 0, a + c < b + c.
Multiplication Compatibility Property
- For any three integers a, b, and c, if a < b and c > 0, then a · c < b · c.
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