Addition and multiplication on Z, Q, R and C are both commutative.

Subtraction on these sets is not commutative. For example, 3 - 2 is not the same as 2 - 3.

Matrix multiplication (of n by n matrices) is not commutative if n is at least 2. For example,

[[0,1],[0,0]] · [[1,0],[0,0]] = [[0,0],[0,0]],

whereas

[[1,0],[0,0]] · [[0,1],[0,0]] = [[0,1],[0,0]].

• Composition of functions from R to R is not commutative. For instance f(x) = x² and g(x) = x + 1 do not commute:
  f(g(x)) = x² + 2x + 1 and g(f(x)) = x² + 1

  are not the same for all values of x (substitute x = 1 for example), so the functions fg and gf are not the same.

• Similarly, the product of permutations is not commutative. For example, the cycles (1,2) and (2,3) do not commute:
  (1,2)(2,3) = (1,2,3), whereas (2,3)(1,2) = (1,3,2).