Seeking explanation of the meaning of translation-invariance for order on the set of integers (Z) | Step-by-Step Solution

Understanding Translation-Invariant Order on ℤ

What We're Solving: We need to understand what it means for the usual order relation on the integers (ℤ) to be "translation-invariant." This is a fundamental property that connects ordering with the algebraic structure of integers.

The Approach: We'll explore what "translation" means in this context, then see how the order behaves when we "translate" (shift) numbers. This will help you understand why this property is so important in abstract algebra.

Step-by-Step Solution:

Step 1: Understanding "Translation" on ℤ In the context of integers, a "translation" means adding the same number to multiple integers. Think of it as sliding numbers along the number line by a fixed distance.

• Example: Translating by +3 moves: 1→4, 5→8, -2→1

Step 2: What Does "Order" Mean? The order on ℤ is our familiar "less than or equal to" relation (≤). We know that 2 < 5, -3 < 0, etc.

Step 3: Putting It Together - Translation-Invariant Order The order is translation-invariant if: whenever we have two integers in a certain order relationship, that same relationship is preserved when we translate both integers by the same amount.

Formally: If a ≤ b, then a + c ≤ b + c for any integer c.

Step 4: Why This Makes Intuitive Sense Think about the number line! If point A is to the left of point B, then sliding both points the same distance in the same direction preserves their relative positions. A will still be to the left of B.

Step 5: Concrete Examples

• We know 3 < 7
• Translation-invariance tells us: 3 + (-2) < 7 + (-2), so 1 < 5 ✓
• And: 3 + 10 < 7 + 10, so 13 < 17 ✓
• This works for any translation!

The Answer: Translation-invariance of order on ℤ means that the relative ordering between any two integers is preserved when we add the same number to both integers. Mathematically: if a ≤ b, then a + c ≤ b + c for all integers a, b, c.

This property is crucial because it shows how the algebraic operation of addition is compatible with the order structure on ℤ.

Memory Tip: Think "sliding preserves order" - imagine two people on a moving walkway. No matter how far the walkway moves them, their relative positions stay the same! The person who was ahead stays ahead.

This concept connects beautifully to the idea of ordered groups in abstract algebra - you're seeing how algebraic structure and order can work together harmoniously!