Order Properties of Integers Quiz

Study Notes

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.

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)
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

For any three integers, a, b, and c, if a < b and b < c, then a < c.
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.

For any two integers, a and b, if a < b, then it is not the case that b < a.
If a ≤ b and b ≤ a, then a = b.
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).

For any three integers a, b, and c, if a < b, then a + c < b + c.
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.