Order Properties of the System (ℤ, +)

Study Notes

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

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.

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.

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.

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.

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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