Commutative law is the most common and first law that is taken over different operands. In general, we have only addition and multiplication operands in different fields such as numbers, matrices, vectors, etc following commutative characteristic. Subtraction and division never follow this characteristic of operands. Some other characteristics of operations are associative law, distributive law, identity law, etc.

The most common use of commutative law is in the calculation part, as we generally interchange positions of numbers or expressions in equations. This is possible only when commutative law holds. Also, the associative law is derived from the commutative law only. According to the associative law, if * is an operand in three variables, x, y and z then,
x * (y * z) = (x * y) * z

## Commutative Law Proof

If * is any operation may it be in sets, in Boolean algebra etc. on two given variables M and N then according to the commutative law

M * N = N * M

If commutative law holds then this condition follows and vice versa.

For example we have 5 + 3 = 8 = 3 + 5

Similarly, 3 x 5 = 5 x 3 = 15

In many other operations also, the commutativity holds true and can be proved easily.

## Commutative Law of Boolean Algebra

In Boolean algebra, there are two common operations-

‘$\wedge$’: This symbol stands for intersection or in simple words multiplication in Boolean variables. This is also known as LOGICAL AND.

‘$\vee$’: This symbol stands for union or in simple words addition in Boolean variables. This is also known as LOGICAL OR.
If a and b are two Boolean variables, then we have commutative law as below

a $\wedge$ b = b $\wedge$ a

a $\vee$ b = b $\vee$ a

That is in both the operations of “logical and” and “logical or”, the commutative law holds true.
For example if we have 0 and 1

Then 0 $\vee$ 1 = 1 and 1 $\vee$ 0 = 1

Also, 0 $\wedge$ 1 = 0 and 1 $\wedge$ 0 = 0

Thus, the commutative law holds true in Boolean algebra.

## Commutative Law of Set

In sets again we have two operations.

$\cup$: This is the symbol for union of two sets

$\cap$: This is the symbol for intersection of two sets.
In set theory, if A and B are two sets, then again the commutative law holds true in both union and intersection of sets, that is,

A $\cup$ B = B $\cup$ A

A $\cap$ B = B $\cap$ A
For example if we have A = {1, 2, 3} and B = {3, 4, 5}

A $\cup$ B = {1, 2, 3, 4, 5} and B $\cup$ A = {1, 2, 3, 4, 5}

Also A $\cap$ B = {3} and B $\cap$ A = {3}

Hence the commutative law holds true in set theory.

We commonly add two whole numbers, integers or fractions; in short, we can say complex numbers. So if x and y are two complex numbers, then commutative law holds true under addition operations of numbers.

That is x + y = y + x for any x and y that are complex numbers.
For example: If we have 5.4 + 6.5 = 11.9

Also 6.5 + 5.4 = 11.9

Hence, the commutative law holds true in addition of numbers.

## Commutative Law in Vector Addition

Vectors are the quantities that have two components: a magnitude which is numeral and a direction which it is pointing. For example: displacement, velocity etc are vectors components; while distance, speed etc are the scalar quantities that are the ones in which there is only a numeric magnitude (can also be expression), in general, no direction only the measurement is given. It is important to note that since vectors have directions associated to them, so the zero vector is also of importance while simply 0 is not. And also, the sign of the vector is also very important.
We represent vectors by an arrow above the variable or vector name.

Vectors can also be added just like numbers, the only difference is we can either add only the magnitudes of the vectors of add them component wise as vectors can be represented in components in xy plane or xyz plane. If a and b are two vectors, then by the commutative law of vector addition

$\vec{a}$ + $\vec{b}$ = $\vec{b}$ + $\vec{a}$
For example: If we are given two vectors a = 4i + 2j – 5k and b = 2i – 5j + 7k, then

a + b = (4 + 2)i + (2 – 5)j + (-5 + 7)k = 6i – 3j + 2k

Also, b + a = (2 + 4)i + (-5 + 2)j + (7 – 5)k = 6i – 3j + 2k

Hence, the commutative law of vector addition proved here as well.