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Commutative Property | introduction with examples

The word commutative comes from commute or to move around. Therefore, we can define commutative property as something that has the tendency of moving stuff around. This property can be used in different ways in order to explain it. In case of addition, the rule is: a + b = b + a while in form numbers, it means 2 + 3 = 3 + 2. Furthermore, in case of multiplication, the rule is ab = ba but in actual numerical forms, it means: 2×3 = 3×2

Wherever commutative property is used; it means they want you to move stuff around. Therefore, once any computation solely depends on moving stuff around, you are required to say that such computation uses the commutative property. An example is given below:

  • With the use the commutative property, restate 3 × 4 × x in at least two ways
  • This means you should move the above equation around and not to simplify it and it can be one of this: 4 × 3 × x or 3 × x × 4 or x × 3 × 4 or x × 4 × 3or 4 × x × 3

The commutative property is any property that is generally connected with binary operations as well as functions. If the commutative property holds for any pair of elements under a particular binary operation, then the 2 elements are considered to commute under such operation

In addition to the above explanations, we also have commutative property in our everyday activities. The example of such activities is given below:

    Wearing of socks
  • The commutativity of addition

Wearing of socks: the putting on of socks also resembles a commutative property. This is true because irrespective of which sock you put on first or which leg you wear them on first does not make a difference so long as you put on socks

The commutativity of addition: this can be seen especially while trying to pay for a particular item. Regardless of the methods of payment, your bills will still be the same. This is an example of commutative property

Commutative property can also be found in some mathematical operations such as in the addition of vectors. There are 2 known examples of commutative binary operations:

  • The addition of real numbers, e.g. 5 + 4 = 4 + 5, in both cases, our final answer still equals 9
  • The multiplication of real numbers is commutative, e.g. 5 × 3 = 3 × 5 in both situations, the final answer will still be 15

Some of the binary truth functions are as well of commutative property. Since the truth tables for these functions are similar when one of them changes the order of the operands. For instance, the logical biconditional function p ↔ q is the same as q ↔ p. This function is also written as p IFF q, or also written as p ≡ q, and can be written like this Epq. This last form of expressing the commutative property is one of the examples of the most concise notation in the article on truth functions which lists the 16 possible binary truth functions. Among these 16 possible functions, only 8 are said to be commutative property:

  • Vpq = Vqp
  • Apq (OR) = Aqp
  • Dpq (NAND) = Dqp
  • Epq (IFF) = Eqp
  • Jpq = Jqp
  • Kpq (AND) = Kqp
  • Xpq (NOR) = Xqp
  • Opq = Oqp

Further commutative property in binary operations include the following examples:

  • Addition as well as multiplication of complex numbers as a commutative property form
  • Addition and scalar multiplication of vectors as a form of commutative property
  • intersection and union of sets as a form of commutative property

History of cumulative property and everyday life cumulative operations

The first known use of the term cumulative property was in a French journal that was published in 1814. Records of the use of the commutative property have been dated back as far as to the ancient times. The Egyptians used it as a means of multiplication to simplify computing products

Euclid is known to have assumed the commutative property of multiplication in his book as well. Formal uses of the commutative property came to play in the late 18th and also in the early 19th centuries when mathematicians began to work on a theory of functions. Today it is a well-known and basic property mostly used in some branches of mathematics

The first recorded use of the commutative term was in a memoir by François Servois in 1814 who used the word commutatives while trying to describe functions that possess what is now known as the commutative property. The word is basically the use of a French word commuter, which means to change or to switch and the suffix - ative - meaning, tending to. So the word literally means tending to substitute or switch

There is also non-commutative property in our everyday life, among which are as follows:

  • Concatenation is the act of joining character strings together, is a noncommutative operation. For example: EA + T = EAT ≠ TEA = T + EA. If it was a form of commutative property, it will definitely be the same result
  • Washing and drying clothes also resemble a non-commutative operation because washing and then drying gives a different end point to drying and then washing. In case of commutative property, the end result never changes
  • Rotating a book 90° round the y-axis then 90° around the x-axis produces a different orientation than when the rotations are performed in the opposite order, but if it was in a commutative property, the result will be the same even after changing the axis of rotations
  • The twists of the Rubik's cube are as well noncommutative. This can be studied using group theory
  • Thought processes are also noncommutative. For instance, a person asked a question about kites but then, another question about the sky might give a different answer to the question about sky rather than an answer that should be similar to the question about a kite and this is because the person’s state of mind might change while asking a particular question. This is not so in case of a commutative property

Some other non-commutative binary operations found in the field of mathematics include the below list:

  • Subtraction is noncommutative since 0 – 1 ≠ 1 – 0, it would have been the same result if it was commutative property
  • Division is also non-commutative, since 12 ≠ 21 is not part of commutative property
  • Likewise, some truth functions are non-commutative as well, since the truth tables for the functions are different when one changes the order of the operations unlike in a commutative property. For example; the truth tables for f (A, B) = A Λ ¬B (A AND NOT B) and f(B, A) = B Λ ¬A can be represented in the table below:

     

    A

     

    B

     

    f (A, B)

     

    f (B, A)

     

    F

     

    F

     

    F

     

    F

     

    F

     

    T

     

    F

     

    T

     

    T

     

    F

     

    T

     

    F

     

    T

     

    T

     

    F

     

    F

     

  • Matrix multiplication is noncommutative since:

     

    [0 2 0 1 ]= [1 1 0 1 ]. [0 1 0 1 ] ≠ [0 1 0 1 ]. [1 1 0 1 ]= [0 1 0 1 ]
  • The vector product otherwise called the cross product of 2 vectors in 3 dimensions is anti-commutative, i.e., b × a = − (a × b)

Another closely related properties to the commutative property is the associative property. This kind of property contains 2 or more occurrences of the same operator states such that the order operations are performed but does not affect the end result as long as the order of terms will not change. Well, it is not like the commutative property in which the end results are affected always. Furthermore, most of the commutative operations encountered in practice are as well associative

There are some forms of symmetry in the field of mathematics and can be linked directly to commutativity. When a commutative property is written as a binary function then the resulting function is symmetric across the line y = x. e.g. if we allow a function f denote addition so that f(x, y) = x + y, then f will become a symmetric function

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