Commutative operations

Commutative operations . In mathematics , some mathematical operations have the commutative property or commutativity in which the result does not vary when changing the order of the elements on which it is applied. This property is true, for example, in the addition and multiplication of real numbers: the order of the addends does not alter the sum and the order of the factors does not alter the product .


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  • 1 Definition
  • 2 Examples
  • 3 Various cases
  • 4
  • 5 References


In general, given a group (A, *) , the internal operation * is commutative if for any pair of elements a and b of A it is true that a * b = b * a .

Let the pair be <S; *> where S is a non-empty set and * an operation on S, if for two of its elements a and b it is true that a * b = b * a , * is said to be a commutative operation on S.


  • In real numbers, addition and multiplication are commutative operations. For example, 2 + 1 = 3 = 1 + 2 and 3 · 2 = 6 = 2 · 3.
  • In complex numbers, addition and multiplication are commutative operations. For example, (1 + i) – 2i = 1 – i = – 2i + (1 + i).
  • In the group of real square matrices, the sum is a commutative operation but the product is not.
  • The composition of functions is not a commutative operation.
  • The vector product of vectors in R 3is not commutative.

Various cases

In propositional logic

  1. The conjunction p and qis the same as the proposition q and p .
  2. the inclusive disjunction because both are thesame as both
  3. the exclusive disjunction for only oneis the same as for only one .
  4. the double implication p if only qis the same as q if only if

The commutativity is verified for each operation by means of a table of values [1]

In set theory

  1. The union of sets A union B = B union A
  2. the intersection of sets A inter B = B inter A
  3. the symmetric difference A Δ B = B Δ A

The set of the first member is shown to be part of the set of the second member and vice versa, using the logical propositions of the case. [2]

In numerical systems

  1. In natural numbers: it is true that a + b = b + a, is demonstrated with the Peano axiomatic. [3]
  2. In the integers. Let M = (a, b) and N = (c, d). Then M + N = (a + c, b + d) and N + M = (c + a, d + c) applying the additive commutativity of the naturals on the components of the representative ordered pairs the commutative property is fulfilled.
  3. In the rational numbers L = (m, n) and H = (p, q). We have L + H = (mq + np, nq); H + L = (pn + mq, qn), as in the ordered pairs there are integers that fulfill the commutative property of addition and multiplication.
  4. In real numbers. Let a, b, c and d be any rational numbers such that a≤x≤b, c≤y≤d where x and y are real numbers, the sum is such that a + c≤x + y≤b + d. This definition tests the commutative property. [4]
  5. adding complex numbers is a commutative operation.
  6. the addition of quaterniums is a commutative operation.

In reverse operations

  1. the subtractionin the integer, rational, real, complex, quaternion is an inverse operation of addition, but is not commutative, except for aa. This occurs because the subtraction solves the equation a + x = b and this does not equal the equation b + x = a, which would ensure the commutative property of the subtraction.
  2. the division of numbers is not commutative except in the b / b and non-null case. Division solves the equation bx = a, it does not solve ax = b at the same time. therefore two equations fit: bx = a, plus ay = b. [5]

In algebra

  1. The sum of polynomials in an indeterminate is commutative P (x) + Q (x) = Q (x) + P (x)
  2. The sum of matrices and that of vectors of the same dimension is commutative.
  3. The dot product of two vectors is commutative
  4. The vector product of two vectors is not commutative
  5. The product of two permutations is not commutative
  6. The matrix product is not commutative [6]
  7. The quaternium product is not commutative [7]

Algebraic systems

  1. The operation that defines the group is not necessarily commutative; in case it is an abelian group.
  2. A ring is an algebraic system that involves two operations. The first called addition that defines the abelian group with the elements of the ring.
  3. A body is a ring in which each element has a multiplicative inverse. If the body is commutative with respect to multiplication it is called a field[8]
  4. Both the R-module A and the K-vector space V, with respect to the addition of its elements, are a commutative group [9].
  5. In an algebraic group a [x, y] = xyx -1-1 is called a commutator of x with y [10] . The set of all switches of group G, form a subgroup of G.


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