A binary operation * defined on `Z^(+)` defined as a * b = `2^(ab)`. Determine whether * is associative

Answer

To determine whether the binary operation * defined on \( \mathbb{Z}^+ \) (the set of positive integers) as \( a * b = 2^{ab} \) is associative, we need to check if the following condition holds for all \( a, b, c \in \mathbb{Z}^+ \):

\[ a * (b * c) = (a * b) * c \]

### Step 1: Calculate \( b * c \)

Using the definition of the operation, we have:

\[ b * c = 2^{bc} \]

### Step 2: Calculate \( a * (b * c) \)

Now, substituting \( b * c \) into the operation with \( a \):

\[ a * (b * c) = a * (2^{bc}) = 2^{a \cdot 2^{bc}} \]

### Step 3: Calculate \( a * b \)

Next, we calculate \( a * b \):

\[ a * b = 2^{ab} \]

### Step 4: Calculate \( (a * b) * c \)

Now, substituting \( a * b \) into the operation with \( c \):

\[ (a * b) * c = (2^{ab}) * c = 2^{(2^{ab})c} \]

### Step 5: Compare \( a * (b * c) \) and \( (a * b) * c \)

We need to compare the two results we obtained:

1. \( a * (b * c) = 2^{a \cdot 2^{bc}} \)
2. \( (a * b) * c = 2^{(2^{ab})c} \)

To check if these two expressions are equal, we can analyze them further:

- The left-hand side is \( 2^{a \cdot 2^{bc}} \).
- The right-hand side is \( 2^{(2^{ab})c} \).

### Step 6: Simplifying the expressions

Let's simplify both sides:

1. The left-hand side can be expressed as \( 2^{a \cdot 2^{bc}} \).
2. The right-hand side can be expressed as \( 2^{2^{ab} \cdot c} \).

For the two expressions to be equal, we need:

\[ a \cdot 2^{bc} = 2^{ab} \cdot c \]

### Step 7: Conclusion

Since \( a \cdot 2^{bc} \) and \( 2^{ab} \cdot c \) are not generally equal for all positive integers \( a, b, c \), we conclude that:

\[ a * (b * c) \neq (a * b) * c \]

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