Commutative vs. Associative — What's the Difference?

Relating to, involving, or characterized by substitution, interchange, or exchange.

Of, characterized by, resulting from, or causing association.

Independent of order. Used of a logical or mathematical operation that combines objects or sets of objects two at a time. If a × b = b × a, the operation indicated by × is commutative.

(Mathematics) Independent of the grouping of elements. For example, if a + (b + c) = (a + b) + c, the operation indicated by + is associative.

Such that the order in which the operands are taken does not affect their image under the operation.

Pertaining to, resulting from, or characterised by association; capable of associating; tending to associate or unite.

Having a commutative operation.

Such that, for any operands $a,\; b$ and $c$, $(a\; *\; b)\; *\; c\; =\; a\; *\; (b\; *\; c)$; (of a ring, etc.) whose multiplication operation is associative.

Such that any two sequences of morphisms with the same initial and final positions compose to the same morphism.

(computing) Addressable by a key more complex than an integer index.

Relating to exchange; interchangeable.

Having the quality of associating; tending or leading to association; as, the associative faculty.

Relative to exchange; interchangeable; reciprocal.

Relating to or resulting from association;

Having the property of commutativity.

Characterized by or causing or resulting from association;

Of a binary operation; independent of order; as in e.g.