** Associative vs Commutative **

In our day to day lives, we have to use numbers whenever we need to get a measure of something. At the grocery store, at the gas station, and even in the kitchen, we need to add, subtract, and multiply two or more quantities. From our practice, we perform these calculations quite effortlessly. We never notice or question why we do these operations in this particular way. Or why these calculations can’t be done in a different way. The answer is hidden in the way these operations are defined in the mathematical field of algebra.

In algebra, an operation involving two quantities (such as addition) is defined as a binary operation. More precisely it is an operation between two elements from a set and these elements are called the ‘operand’. Many operations in mathematics including arithmetic operations mentioned earlier and the ones encountered in the set theory, linear algebra, and mathematical logic can be defined as binary operations.

There is a set of governing rules pertaining to a specific binary operation. Associative and the commutative properties are two fundamental properties of the binary operations.

**More about Commutative Property**

Suppose some binary operation, denoted by the symbol ⊗, is performed on the elements *A* and *B*. If the order of the operands is not affecting the result of the operation, then the operation is said to be commutative. i.e. if *A *⊗* B *=* B *⊗* A* then the operation is commutative.

The arithmetic operations addition and multiplication are commutative. The order of the numbers added together or multiplied together does not affect the final answer:

*A* + *B* = *B* + *A* ⇒ 4 + 5 = 5 + 4 = 9

*A* × *B *= *B* × *A* ⇒ 4 × 5 = 5 × 4 = 20

But in the case of division change in the order gives the reciprocal of the other, and in subtraction the change give the negative of the other. Therefore,

*A *- *B *≠ *B *- *A* ⇒ 4 – 5 = -1 and 5 – 4 = 1

*A *÷ *B *≠ *B *÷ *A* ⇒ 4 ÷ 5 = 0.8 and 5 ÷ 4 = 1.25 [in this case *A*,*B *≠ 1 and 0 ]

In fact, the subtraction is said to be anti-commutative; where *A *- *B *= – (*B *- *A*).

Also, the logical connectives, the conjunction, disjunction, implication, and the equivalence, also are commutative. Truth functions are also commutative. The set operations union and intersection are commutative. Addition and the scalar product of the vectors are also commutative.

But the vector subtraction and vector product is not commutative (vector product of two vectors is anti-commutative). The matrix addition is commutative, but the multiplication and the subtraction are not commutative. (Multiplication of two matrices can be commutative in special cases, such as the multiplication of a matrix with its inverse or the identity matrix; but definitely matrices are not commutative if the matrices are not of the same size)

**More about Associative Property**

A binary operation is said to be associative if the order of the execution does not affect the result when two or more occurrences of the operator is present. Consider the elements *A, B* and *C* and the binary operation ⊗ . The operation ⊗ is said to be associative if

*A *⊗ *B *⊗ *C *= *A *⊗ (*B *⊗ *C*) = (*A *⊗ *B*) ⊗ *C*

From the basic arithmetic functions, only addition and the multiplication are associative.

*A *+ (*B *+ *C*) = (*A *+ *B*) + *C* ⇒ 4 + (5 + 3) = (5 + 4) + 3 = 12

*A *× (*B *× *C*) = (*A *× *B*) × *C* ⇒ 4 × (5 × 3) = (5 × 4) ×3 = 60

The subtraction and division are not associative;

*A *- (*B *- *C*) ≠ (*A *- *B*) – *C* ⇒ 4 – (5 – 3) = 2 and (5 – 4) – 3 = -2

*A *÷ (*B *÷ *C*) ≠ (*A *÷ *B*) ÷ *C* ⇒ 4 ÷ (5 ÷ 3) = 2.4 and (5 ÷ 4) ÷ 3 = 0.2666

The logical connectives disjunction, conjunction, and equivalence are associative, as also the set operations union and intersection. The matrix and vector addition are associative. The scalar product of vectors is associative, but the vector product is not. Matrix multiplication is associative only under special circumstances.

**What is the difference between Commutative and Associative Property?**

• Both associative property and the commutative property are special properties of the binary operations, and some satisfies them and some do not.

• These properties can be seen in many forms of algebraic operations and other binary operations in mathematics, such as the intersection and union in set theory or the logical connectives.

• The difference between commutative and associative is that commutative property states that the order of the elements does not change the final result while associative property states, that the order in which the operation is performed, is not affecting the final answer.

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