The binary operation defined on the set z of all integers as `a ** b = |a-b| - 1` is

Options

A. Commutative
B. Associative
C. Non - associative
D. Non - commutative

Answer

To determine the properties of the binary operation defined on the set of all integers \( \mathbb{Z} \) as \( a \star b = |a - b| - 1 \), we will check if it is commutative, associative, or neither.

### Step 1: Check for Commutativity
A binary operation \( \star \) is commutative if \( a \star b = b \star a \) for all \( a, b \in \mathbb{Z} \).

1. Calculate \( a \star b \):
\[ a \star b = |a - b| - 1 \]

2. Calculate \( b \star a \):
\[ b \star a = |b - a| - 1 \]

3. Since \( |a - b| = |b - a| \), we have:
\[ b \star a = |b - a| - 1 = |a - b| - 1 = a \star b \]

Thus, \( a \star b = b \star a \), which shows that the operation is **commutative**.

### Step 2: Check for Associativity
A binary operation \( \star \) is associative if \( (a \star b) \star c = a \star (b \star c) \) for all \( a, b, c \in \mathbb{Z} \).

1. Calculate \( (a \star b) \star c \):
- First, find \( a \star b \):
\[ a \star b = |a - b| - 1 \]
- Let \( d = a \star b = |a - b| - 1 \). Now calculate \( d \star c \):
\[ d \star c = |d - c| - 1 = ||a - b| - 1 - c| - 1 \]

2. Calculate \( a \star (b \star c) \):
- First, find \( b \star c \):
\[ b \star c = |b - c| - 1 \]
- Now calculate \( a \star (b \star c) \):
\[ a \star (b \star c) = |a - (|b - c| - 1)| - 1 = |a - |b - c| + 1| - 1 \]

3. To check for equality, we need to compare \( ||a - b| - 1 - c| - 1 \) and \( |a - |b - c| + 1| - 1 \).

4. Let's take specific values to test associativity:
- Let \( a = 2, b = 3, c = 5 \):
- Calculate \( (a \star b) \star c \):
\[ a \star b = |2 - 3| - 1 = 1 - 1 = 0 \]
\[ 0 \star 5 = |0 - 5| - 1 = 5 - 1 = 4 \]
- Calculate \( a \star (b \star c) \):
\[ b \star c = |3 - 5| - 1 = 2 - 1 = 1 \]
\[ 2 \star 1 = |2 - 1| - 1 = 1 - 1 = 0 \]

5. Since \( (a \star b) \star c = 4 \) and \( a \star (b \star c) = 0 \), we conclude:
\[ (a \star b) \star c \neq a \star (b \star c) \]

Thus, the operation is **not associative**.

### Conclusion
The binary operation defined as \( a \star b = |a - b| - 1 \) is **commutative** but **not associative**.