Let A be any non zero set more than one elements * is a binary operation on A defined as a* b= ` a AA a,b in A ` . Check the commutativity and associativity.

Answer

To solve the problem, we need to check the properties of commutativity and associativity for the binary operation defined on the set \( A \).

### Step 1: Define the operation
We are given a binary operation \( * \) on a non-zero set \( A \) with more than one element, defined as:
\[ a * b = a \text{ for } a, b \in A \]

### Step 2: Check for commutativity
To check if the operation is commutative, we need to verify if:
\[ a * b = b * a \]
Using the definition of the operation:
\[ a * b = a \quad \text{and} \quad b * a = b \]
For commutativity to hold, we need:
\[ a = b \]
However, since \( a \) and \( b \) are elements of \( A \) and \( A \) has more than one element, \( a \) is not necessarily equal to \( b \). Therefore:
\[ a * b \neq b * a \]
Thus, the operation \( * \) is **not commutative**.

### Step 3: Check for associativity
To check if the operation is associative, we need to verify if:
\[ (a * b) * c = a * (b * c) \]
Using the definition of the operation:
1. Calculate \( a * b \):
\[ a * b = a \]
2. Now calculate \( (a * b) * c \):
\[ (a * b) * c = a * c = a \]
3. Now calculate \( b * c \):
\[ b * c = b \]
4. Now calculate \( a * (b * c) \):
\[ a * (b * c) = a * b = a \]

Now we compare both sides:
\[ (a * b) * c = a \quad \text{and} \quad a * (b * c) = a \]
Since both sides are equal, we conclude that:
\[ (a * b) * c = a * (b * c) \]
Thus, the operation \( * \) is **associative**.

### Summary of Results
- The operation \( * \) is **not commutative**.
- The operation \( * \) is **associative**.