# Commutative property

The commutative property does not imply the associative property, nor vice versa. For example, matrix multiplication is associative but not commutative. Also, define the function $\Box$ on integers by $a \Box b = 2(a + b)$. Although, $\Box$ is commutative, it is not associative, as $((1 \Box 0) \Box 0) = 2 \Box 0 = 4$ but $(1 \Box (0 \Box 0)) = 1 \Box 0 = 2$.