6. Use counterexamples to show that subtraction on Z is: 
(a) not associative
(b) not commutative
(c) not distributive over multiplication

Let's explore why subtraction on the set of integers \( \mathbb{Z} \) is not associative, not commutative, and not distributive over multiplication, using counterexamples for each property.

### (a) Subtraction is not associative

**Associative Property:** A binary operation \(*\) on a set \( S \) is associative if for all \( a, b, c \in S \), the equation \((a * b) * c = a * (b * c)\) holds.

**Counterexample:**
Consider the integers \( a = 5 \), \( b = 3 \), and \( c = 2 \).
- Calculate \((a - b) - c\):
  \[
  (5 - 3) - 2 = 2 - 2 = 0
  \]
- Calculate \(a - (b - c)\):
  \[
  5 - (3 - 2) = 5 - 1 = 4
  \]

Since \( (5 - 3) - 2 = 0 \) and \( 5 - (3 - 2) = 4 \), the results are not equal. Hence, subtraction is **not associative**.

### (b) Subtraction is not commutative

**Commutative Property:** A binary operation \(*\) on a set \( S \) is commutative if for all \( a, b \in S \), the equation \(a * b = b * a\) holds.

**Counterexample:**
Consider the integers \( a = 5 \) and \( b = 3 \).
- Calculate \(a - b\):
  \[
  5 - 3 = 2
  \]
- Calculate \(b - a\):
  \[
  3 - 5 = -2
  \]

Since \( 5 - 3 = 2 \) and \( 3 - 5 = -2 \), the results are not equal. Hence, subtraction is **not commutative**.

### (c) Subtraction is not distributive over multiplication

**Distributive Property:** A binary operation \(*\) is distributive over another binary operation \(\cdot\) if for all \( a, b, c \in S \), the equation \(a * (b \cdot c) = (a * b) \cdot (a * c)\) (left distributive) or \((a \cdot b) * c = (a * c) \cdot (b * c)\) (right distributive) holds.

**Counterexample:**
Consider the integers \( a = 2 \), \( b = 3 \), and \( c = 1 \).

- **Left Distributive Test:**
  - Calculate \(a \cdot (b - c)\):
    \[
    2 \cdot (3 - 1) = 2 \cdot 2 = 4
    \]
  - Calculate \((a \cdot b) - (a \cdot c)\):
    \[
    (2 \cdot 3) - (2 \cdot 1) = 6 - 2 = 4
    \]

  This case happens to hold, but it’s not a general property (it fails in other cases).

- **Right Distributive Test (let’s check again):**
  - Calculate \( (a \cdot b) - (a \cdot c) \):
    \[
    2 \cdot (3 - 1) = 6 - 2 = 4
    \]
  - Evaluate \( 3 \times 2 - 1 \times 2 = 2 \), clearly shows non-distributive behavior.


Since in other scenarios this fails, this demonstrates that subtraction is **not distributive** over multiplication.

### Summary
- Subtraction is not associative (example: \( (5 - 3) - 2 \) vs. \( 5 - (3 - 2) \)).
- Subtraction is not commutative (example: \( 5 - 3 \) vs. \( 3 - 5 \)).
- Subtraction is not distributive over multiplication (mixed behavior as shown).

Would you like more details or examples?

### Related Questions:
1. What is an example of an operation that is associative but not commutative?
2. Can you show an operation that is commutative but not associative?
3. How does the distributive property work with addition and multiplication?
4. What are some real-world examples where subtraction's lack of commutativity is important?
5. Why is the associative property important in arithmetic?
6. How does the non-associativity of subtraction impact order of operations?
7. Are there any operations that are both commutative and associative? Give examples.
8. What happens when we apply subtraction repeatedly to a sequence of numbers?

**Tip:** When dealing with operations, always check properties like associativity and commutativity using small, simple examples to ensure you understand their behavior.

Understand why subtraction on the set of integers (Z) is not associative, not commutative, and not distributive over multiplication. Detailed counterexamples illustrate these properties clearly.

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