Commutative Law

Commutative law only applies to addition and multiplication operations in mathematics. It does not, however, apply to the other two mathematical operations, subtraction and division. If a and b are any two integers, the addition and multiplication of a and b produce the same result regardless of the position of a and b, according to commutative law or commutative property. It can be represented symbolically as:

                                           A + B = B + A

                                           A × B = B × A

For example let us consider two numbers 4 and 7.

So we can easily say that ;

              4 + 7 = 7 + 4

              4 × 7 = 7 × 4

Definition:-

The commutative law asserts that when two numbers are added or multiplied, the final value remains the same regardless of the position of the two numbers. To put it another way, the order in which two real numbers are added or multiplied has no bearing on the result.

As a result, if P and Q are two real numbers, then according to this law;

                              P + Q = Q + P

                              P × Q = Q × P

Subtraction and division are not covered by commutative law.

Proof:-

For addition and multiplication, proving the commutative property is simple. Let us use some examples to demonstrate it.

The Commutative Law for Addition:

The commutative law of addition asserts that when two numbers are added, the result is the same as adding the numbers in their reverse order.

                i.e., A + B = B + A

              Examples:-

• 1 + 7 = 7 + 1
• 5 + 9 = 9 + 5
• 3 + (-5) = (-5) + 3

When the first number is negative and the position is changed, the first number’s sign is changed to positive, resulting in:

              P – (-Q) = P + Q ————- (1)

Again, (-Q) – P = -Q – P —————(2)

Clearly from equation (1) &(2)

        P – (-Q) ≠ (-Q) – P

For example: 3 – (-6) ≠ (-6) – 3 

Hence, subtraction is not commutative.

The Commutative Law for multiplication:-

According to this law, the outcome of multiplying two integers remains the same even if the numbers’ places are swapped.

                       i.e., P × Q = Q × P

For example: 

• 1 × 6 = 6 × 1
• 5 × 3 = 3 × 5
• 2 × (-9) = (-9) × 2

The Commutative Law for Percentage:-

When calculating percentages, we can interchange or vary the order of the values, but the result remains the same. On a mathematical level, we can say:

A% of B = B% of A

Example:

Ten percent of fifty equals fifty percent of ten

Since,

20% of 80 =  (20/100) x 80 = 16

80% of 20 = (80/100) x 20 = 16

As a result, the answer remains unchanged.

Associative and Distributive Law:-

Laws of Associative and Distributive Association

Aside from the commutative law, there are two additional major laws that are often utilized in mathematics:

• Associative law
• Distributive law

Associative Law: A, B, and C are three real numbers, then according to this law,

(A + B) + C = A + (B + C)

(A × B) × C = A × (B × C)

This law, like the commutative rule, applies to addition and multiplication.

For example: If 5,7 and 9 are three numbers then;

    5+(7+9) = (5+7)+9

⇒5+16 = 12 + 9

⇒21 = 21

&

     5.(7.9) = (5.7).9

⇒ 5.(63) = (35).9

⇒ 315 = 315

Hence, proved.

Distributive Law: This law is not related to commutative or associative law in any way. If A, B, and C are three real numbers, then, according to this rule:

    A × (B + C) = (A × B) + (A × C)

For example: If 5,7 and 9 are three numbers then;

     5.(7 + 9) = 5.7 + 5.9

⇒ 5.(16) = 35 + 45

⇒ 80 = 80

Hence, proved.

Commutative Law for sets:

A grouping of goods or objects is referred to as a “set.” Intersection, union, and difference are just a few of the operations that may be performed on sets that we’ve learned about.

According to the Commutative Law for Union of Sets and the Commutative Law for Intersection of Sets, the order in which the operations are done has no bearing on the outcome.

If P and Q are two different sets, then, as per commutative law;

P ∪ Q = Q ∪ P  [Union of sets]

P ∩ Q = Q ∩ P   [Intersection of sets]

For example, if A = {4, 5, 6} and B = {6, 7, 8, 9}, then;

A Union B = A ∪ B = {4, 5, 6, 7, 8, 9} ————- (1)

B Union A = B ∪ A = {4, 5, 6, 7, 8, 9} ————- (2)

From (1) and (2), we get;

                      A ∪ B = B ∪ A

Now, 

A intersection B = A ∩ B = {6}  ——— (3)

B intersection A = B ∩ A = {6}  ——— (4)

From (3) and (4), we get;

                     A ∩ B = B ∩ A

As a result, commutative law for union and intersection of two sets was established.

Conclusion:-

In math, we examined the commutative property, which states that “changing the order of the operands has no effect on the output.” The commutative property has only been demonstrated to apply to multiplication and addition. Subtraction and division, on the other hand, do not follow the commutative property.