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:
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Left-hand side: \[ (a * b) * c = (a - b) * c = (a - b) - c = a - b - c \]
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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 \( \boxed{5} \).