An internal operation ❄︎ on a set E is associative if, for all elements a, b and c in E, we have : (a ❄︎ b) ❄︎ c = a ❄︎ (b ❄︎ c).
Examples
The addition and multiplication of real numbers are associative operations.
- (12 + 14) + 16 = 12 + (14 + 16)
- (6 × 5) × 3 = 6 × (5 × 3)
- (x + y) + z = x + (y + z)
- (x × y) × z = x × (y × z)
The subtraction of real numbers is not an associative property.
- (15 – 10) – 3 ≠ 15 – (10 – 3), since : 2 ≠ 8