Commutative Law

In Mathematics, commutative law is applicable only for addition and multiplication operations. But, it is not applied to other two arithmetic operations, such as subtraction and division. As per commutative law or commutative property, if a and b are any two integers, then the addition and multiplication of a and b result in the same answer even if we change the position of a and b. Symbolically it may be represented as:

a+b=b+a

a×b=b×a

For example, if 2 and 5 are the two numbers, then;

2 + 5 = 5 + 2 = 7

2 × 5 = 5 × 2 = 10

Definition

The definition of commutative law states that when we add or multiply two numbers then the resultant value remains the same, even if we change the position of the two numbers. Or we can say, the order in which we add or multiply any two real numbers does not change the result.

Hence, if A and B are two real numbers, then, as per this law;

Note: Commutative law does not work for subtraction and division

Proof

It is easy to prove the commutative law for addition and multiplication. Let prove with examples.

Commutative Law of Addition

The commutative law of addition states that if two numbers are added, then the result is equal to the addition of their interchanged position.

A+B = B+A

Examples:

• 1+2 = 2+1 = 3
• 4+5 = 5+4 = 9
• -3+6 = 6+(-3) = 6-3 = 3

This law is not applicable for subtraction because if the first number is negative and if we change the position, then the sign of the first number will get changed to positive, such that;

(-A)-B = -A – B ……(1)

After changing the position of the first and second number, we get;

B – (-A) = B + A ….(2)

Hence, from equation 1 and 2, we can see that;

(-A)-B ≠ B-(-A)

For example: (-9)-2 = -9-2 = -11

& 2-(-9) = 2+9 = 11

Hence, -11 ≠ 11.

Commutative Law of Multiplication

As per this law, the result of the multiplication of two numbers stays the same, even if the positions of the numbers are interchanged.

Hence, A.B = B.A

Examples:

• 1×2 = 2×1 = 2
• 4×5 = 5×4 = 20
• -3×6 = 6×(-3) = -18

Commutative Law in Percentages

While finding the percentages, if we interchange or swap the order of the values, then the answer does not change. Mathematically we can say;

A% of B = B% of A

Example:

10% of 50 = 50% of 10

Since, 

10% of 50 = (10/100) x 50 = 5

50% of 10 = (50/100) x 10 = 5

Thus, the answer remains the same.

Associative and Distributive Laws

Apart from commutative law, there are other two major laws, which are commonly used in Maths, they are:

• Associative law
• Distributive law

Associative Law: As per this law, if A, B and C are three real numbers, then;

• A+(B+C) = (A+B)+C
• A.(B.C) = (A.B).C

Just like commutative rule, this law is also applicable to addition and multiplication.

For example: If 2,3 and 5 are three numbers then;

2+(3+5) = (2+3)+5

⇒2+8 = 5 + 5

⇒10 = 10

&

2.(3.5) = (2.3).5

⇒ 2.(15) = (6).5

⇒ 30 = 30

Hence, proved.

Distributive Law: This law is completely different from commutative and associative law. According to this law, if A, B and C are three real numbers, then;

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

For example: If 2,3 and 5 are three numbers then;

2.(3+5) = 2.3+2.5

2.(8) = 6+10

16 = 16

Also, read:

• Associative Property
• Distributive Properties

Commutative Law of Sets

Sets are the collection of elements or objects. In sets, we have learned about different types of operations performed on them such as intersection of sets, union of sets, difference of sets, etc.

According to the Commutative law for Union of sets and the Commutative law for Intersection of sets, the order of the sets in which the operations are done, does not change the result.

So, if A and B are two different sets, then, as per commutative law;

A ∪ B = B ∪ A  [Union of sets]

A ∩  B = B ∩ A   [Intersection of sets]

For example, if A = {1, 2, 3} and B = {3, 4, 5, 6}, then;

A Union B = A ∪ B = {1, 2, 3, 4, 5, 6}  …….. (i)

B Union A = B ∪ A = {1, 2, 3, 4, 5, 6}   ……… (ii)

From (i) and (ii), we get;

A ∪ B = B ∪ A

Now, 

A intersection B = A ∩ B = {3}   ……..(iii)

B intersection A = B ∩ A = {3}   ……..(iv)

From (iii) and (iv), we get;

A ∩ B = B ∩ A

Hence, proved commutative law for union and intersection of two sets.