Question

Define a binary operation* on the set of real numbers by a*b= [a-b]. Determine if is associative and/ or commutative and evaluate -1*(-2*4)

Answer 1

To define the binary operation \( * \) on the set of real numbers as \( a * b = a - b \), we can analyze its properties for associativity and commutativity.

### **Commutativity**

A binary operation is said to be commutative if:
\[
a * b = b * a
\]
For our operation:
\[
a * b = a - b
\]
\[
b * a = b - a
\]
Clearly, \( a - b \neq b - a \) unless \( a = b \). Therefore, the operation \( * \) is **not commutative**.

### **Associativity**

A binary operation is associative if:
\[
(a * b) * c = a * (b * c)
\]
Let's compute both sides:

1. **Left-hand side**:
\[
(a * b) * c = (a - b) * c = (a - b) - c = a - b - c
\]

2. **Right-hand side**:
\[
a * (b * c) = a * (b - c) = a - (b - c) = a - b + c
\]

Since \( a - b - c \neq a - b + c \) in general, the operation \( * \) is **not associative**.

### **Evaluation of \(-1 * (-2 * 4)\)**

Now we need to evaluate \(-1 * (-2 * 4)\). First, we compute \(-2 * 4\):
\[
-2 * 4 = -2 - 4 = -6
\]
Now we compute \(-1 * (-6)\):
\[
-1 * (-6) = -1 - (-6) = -1 + 6 = 5
\]

Thus, the result of \(-1 * (-2 * 4)\) is \( \boxed{5} \).