Determine whether the binary operation * on R defined by `aast b =|a-b|` is commutative. Also, find the value of `(-3)^(ast) 2`.

Answer

To determine whether the binary operation * on R defined by \( a * b = |a - b| \) is commutative, we need to check if \( a * b = b * a \) for all real numbers \( a \) and \( b \).

### Step 1: Define the operation
The operation is defined as:
\[ a * b = |a - b| \]

### Step 2: Check commutativity
To check if the operation is commutative, we need to evaluate both \( a * b \) and \( b * a \):
\[ a * b = |a - b| \]
\[ b * a = |b - a| \]

### Step 3: Use properties of absolute values
We know that:
\[ |b - a| = |-(a - b)| = |a - b| \]
This shows that:
\[ b * a = |b - a| = |a - b| = a * b \]

### Conclusion on commutativity
Since \( a * b = b * a \) for all \( a \) and \( b \), we conclude that the operation is commutative.

### Step 4: Calculate the value of \( (-3) * 2 \)
Now, we need to find the value of \( (-3) * 2 \):
\[ (-3) * 2 = |-3 - 2| = |-5| = 5 \]

### Final Answer
Thus, the value of \( (-3) * 2 \) is:
\[ 5 \]

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