Show that the associative property does not hold for integers under subtraction.

Answer

To show that the associative property does not hold for integers under subtraction, we will follow these steps:

### Step 1: Understand the Associative Property
The associative property states that for any three integers \( a \), \( b \), and \( c \):
\[ (a - b) - c \quad \text{should be equal to} \quad a - (b - c) \]
if the property holds.

### Step 2: Set Up the Expression
We will evaluate both sides of the equation using the integers \( a = 7 \), \( b = 5 \), and \( c = 4 \).

### Step 3: Calculate the Left-Hand Side (LHS)
First, we calculate the left-hand side:
\[ LHS = (a - b) - c \]
Substituting the values:
\[ LHS = (7 - 5) - 4 \]
Calculating inside the parentheses first:
\[ = 2 - 4 \]
Now, subtracting:
\[ = -2 \]

### Step 4: Calculate the Right-Hand Side (RHS)
Next, we calculate the right-hand side:
\[ RHS = a - (b - c) \]
Substituting the values:
\[ RHS = 7 - (5 - 4) \]
Calculating inside the parentheses first:
\[ = 7 - 1 \]
Now, subtracting:
\[ = 6 \]

### Step 5: Compare LHS and RHS
Now we compare the results from LHS and RHS:
\[ LHS = -2 \quad \text{and} \quad RHS = 6 \]
Since \( -2 \neq 6 \), we conclude that:
\[ LHS \neq RHS \]

### Conclusion
Since the left-hand side does not equal the right-hand side, we have shown that the associative property does not hold for integers under subtraction.

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