Associative Binary Operations

For binary operation

* : A × A → A

If (a * b) * c = a * (b * c)

Then it is associative binary operation

Addition

+ : R × R → R

* is associative if

  (a * b) * c = a * (b * c)

Since (a * b) * c = a * (b * c) ∀ a, b, c ∈ R

+ is an associative binary operation

Multiplication

× : R × R → R

* is associative if

  (a * b) * c = a * (b * c)

Since (a * b) * c = a * (b * c) ∀ a, b, c ∈ R

 × is an associative binary operation

Subtraction

– : R × R → R

* is associative if

  (a * b) * c = a * (b * c)

Since (a * b) * c ≠ a * (b * c) ∀ a, b, c ∈ R

 –  is not an associative binary operation

Division

÷ : R × R → R

* is associative if

  (a * b) * c = a * (b * c)

Since (a * b) * c ≠ a * (b * c)

÷ is not an associative binary operation