Commutative-property


Commutative property is one of the fundamental math properties. Other important math properties are closure property, associative property and distributive property.
Any operation is said to obey commutative property if a same result is obtained in whatever order the operation is performed.

Among the elementary arithmetic operations of addition, subtraction, multiplication and division, only addition and multiplication obey the commutative property.
i.e. in which ever order the operations of addition and multiplication are carried out, the final result – the sum or product, is the same.

Commutative property on addition:

Addition of numbers is commutative, if a same sum results from any order of operation.

i.e. if a and b are two real numbers, then, if,

a + b = b + a,

then addition is said to obey commutative property.

Example:

2 +3 = 5, and 3 + 2 = 5, so
2 + 3 = 3 + 2

Commutative property on multiplication:

Multiplication of numbers is also commutative, as a same product results from any order of multiplication.

i.e. if a and b are any two numbers, and if
a × b = b × a

then, multiplication is said to obey commutative property.

Example:

2 × 3 = 6, and 3 × 2 = 6, so,
2 × 3 = 3 × 2

Subtraction and Division do not obey commutative property.

Among the four elementary arithmetic operations, the two subtraction and division do not follow the commutative property.

It is because, in both subtraction and division, the order of operation counts.
Unlike in addition and multiplication, where a same sum or product is obtained in whichever order the two operations are carried out, in the case of subtraction and division, the respective results – i.e. the difference and the quotient change on changing the order of the respective operations.

Subtraction is not commutative

Example:

2 + (– 3) = 2 – 3 = -1, and

3 + (– 2) = 3 – 2 = 1

Now, - 1 ≠ 1

So, we can see that the two differences are not same.

Therefore, order of operation matters, and hence, subtraction does not obey commutative property.

Division is not commutative

Example:

2 ÷ 4 = ½, and 4 ÷ 2 = 2,

Now,  ½ ≠ 2,

Since the two quotients are not same, therefore division also does not obey commutative property.

Some other operations which obey the commutative property
1. Union of two sets is commutative.

A U B = B U A

Example:

A = {1, 2, 3} and B = {4, 5, 6},

Then,

A U B = {1, 2, 3, 4, 5, 6} and

B U A = {4, 5, 6, 1, 2, 3}

Since, order in which elements are written in a set does not matter, therefore,

{4, 5, 6, 1, 2, 3} = {1, 2, 3, 4, 5, 6}

Finally,

A U B = B U A

2. Intersection of two sets also obeys commutative property.

3. Matrix addition obeys commutative property.

Distributive property.

Math properties.



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