Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on Q by letting \(x * y = \frac{x+y}{3}\)

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A binary operation * on a set A is a map A × A → A, written (a, b) → a * b. Examples include most of the arthematic operations on the real and complex numbers, such as addition (a+b), multiplication (a × b), subtraction (a – b) and so on.

Given binary operation * defined by x * y = \(\frac{x + y}{3}\) on Rational number (Q)

1) Check if * is commutative

x * y = \(\frac{x + y}{3}\)    ∀ x, y ∈ Q

y * x = \(\frac{y + x}{3}\)  = \(\frac{x + y}{3}\)

Since, x * y = y * x, So, it is commutative

2) Check if * is associative

∀ x, y, z ∈ Q

(x * y) * z = \(\left (\frac{x + y}{3}\right )\) * z = \(\frac{\frac{x + y}{3} + z}{3}\) = \(\frac{x + y + 3z}{9}\)

x * (y * z) = x * \(\left (\frac{y + z}{3}\right )\) = \(\frac{x + \frac{x + y}{3}}{3}\) = \(\frac{3x + y + z}{9}\)

Since, (x * y) * z ≠ x * (y * z), So it is not associative

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