A commutative function $~$f$~$ is a function that takes multiple inputs from a set $~$X$~$ and produces an output that does not depend on the ordering of the inputs. For example, the binary operation $~$+$~$ is commutative, because $~$3 + 4 = 4 + 3.$~$ The string concatenation function concat is not commutative, because concat("3","4")="34" does not equal concat("4","3")="43".