Associative property

Mathematics is neither created nor discovered. It was created from the discovery of the universe

In our description of addition we took most of the examples as the addition of two amounts because anything can be related first between only two things and all other things can be related only after relating first two things in any operation. In the same way addition of any amounts can be added only after adding any two amounts in the given amounts. 

For e.g. to add 1 + 3 + 8 + 10 + 15 + 68 first we will take any two numbers in the given numbers. We can take any two from among these like 1+3 or 1+8 or 8+10 . After adding any two we will add other left numbers following the addition of the first two numbers. In the same way we can add all 5 numbers in 5 times by taking any two numbers taking in order.

This property of adding any two numbers by our own choice made the property called associative.

Hence associative property commonly and simply can be written in the equation form as,

If there are given three numbers a, b, c then from associative property

          a + (b + c) = (a + b) + c

And a, b, c are in whole numbers also can be written as a, b, c ϵ N

The detailed meaning of “ϵ” will be explained later in sets.

This associative property gave another property. In associative property we never mentioned the order of taking numbers. So if there are two numbers then from associative property you can take any two numbers but it is not mentioned in which order. 

So if we add a + b or b + a there is not going to happen any change. We just want pair of sets to add. And it needn’t be in some order in addition. This property is known as commutative property.

It was stated as 

            a + b  =  b + a             and      a, b ϵ N

 

                                                                     

                              

Spontaneously we got two properties which have to be checked in any operation given as associative and commutative property. 

After studying an operation in mathematics we will use that operation in various other mathematical topics. We will apply these operations in all mathematic topics to check whether it is applicable for all the topics or for only the specified topics. That is why the above two properties are so important because we will check it many operations. There are also many properties similar to the above properties.

Many properties like inverse property, associative property, commutative property and other properties will be discussed deeply in binary operations.

Addition also behaves like a property. That is why we are applying addition in all topics like functions, vectors, complex numbers, trigonometry, etc. Addition was used in all these topics.

E.g. Sin (A+B), A vector + B vector, a + ib, f(x) + f(y).