Associative law

Only two of the four major arithmetic operations, addition and multiplication, are covered by the associative law in mathematics. This concept is not applied to other arithmetic operations, such as subtraction and division, because the outcome could change. This is owing to the fact that the position of integers changes during addition and multiplication, but the sign of the numbers does not change.

According to the associative law, if we add or multiply three numbers, their position, order, or arrangement of numbers has no effect on the result. The associative property of addition and multiplication is another name for this law.

What is an associative law? 

The associative law, often known as the associative property, is a mathematical principle that can be used to add and subtract three numbers. Because there will be a change in the result, the associative law can only be used in situations involving addition and multiplication, not subtraction and division. The sign of integers is unaffected by their location in addition and multiplication. As a result, associative law can be used. When it comes to addition and multiplication, the order or position of the numbers has no bearing on the outcome. In vector algebra, the associative property can be utilized to solve vector-related issues.

Associative law formula 

From the definition and examples, the formula for Associative law can be simply established. The associative law is a mathematical rule that can be used to add and multiply three numbers without having to follow any grouping or pattern. With two of the numbers in brackets and one outside, three numbers can be grouped and regrouped, and the problem can be answered starting with the bracketed numbers. Through associative law, grouping or combining numbers in any manner does not provide a different outcome than the original. The three integers can be arranged in the following ways, according to Associative Law:

According to the associative law, grouping variables in an expression has no effect on the outcome.

For the addition of three variables, the associative law is expressed as:

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

For multiplication of three variables, the associative law is expressed as:

A(BC)=(AB)C

 Associative law addition

In mathematics, the addition operation always follows the associative property. That is, the outcomes are independent of the number combination, and any grouping or regrouping will produce the same result. As a result, if the numbers used are x,y and z, addition can be expressed using the associative property in the following fashion.

x+(y+z)=(x+y) +z=x+y+z

 Associative law multiplication

Multiplication, like addition, has an associative property, which means it doesn’t matter how the numbers are grouped. Any relationship or grouping will provide the same outcome. The associative law can be expressed as follows using the numbers x,y and z.

 x×(y×z)=(x×y)×z=x×y×z

 Associative law of addition-proof

The proof of associative law can be established after knowing the formula for associative law. The laws of addition and multiplication are demonstrated in the following sections.

Example:  2+ (3+5) =(2+3)+5  Prove that

2+ (3+5) =2+8=10, LHS first.

Now we can use 

RHS to calculate (2+3) +5=5+5=10

LHS=RHS after comparing both sides of the equation

As a result, it can be proved that

2+(3+5)=(2+3)+5 

Associative law of multiplication-proof

In a similar manner, the evidence of the associative law of multiplication can be provided.

Example: Prove that 2×(3×5)=(2×3)×5 in 

Using the LHS 2(35)=215=30

We can now take RHS,

Following a comparison of both sides of the equation,

LHS=RHS

As a result, it can be proved that

2(35)=(23)5 

(23)5=65=30 

Associative law logic diagram

Figure 1 : Associative law logic diagram

A AND (B AND C) ≡ (A AND B) AND C and then in notation, we’d write:

A Λ (B Λ C) ≡ (A Λ B) Λ C

And also

(A OR B) OR C ≡ A OR (B OR C) and then in notation, we’d write:

(A V B) V C ≡ A V (B V C)

 Conclusion

In this article we learn, rearranging the parenthesis in an expression has no influence on the result due to the associative property of various binary operations in mathematics. In propositional logic, associativity is a valid rule for replacing expressions in logical proofs. The associative law can only be applied to addition and multiplication problems, not subtraction or division. The sign of integers is unaffected by where they are added or multiplied. As a result, it is possible to employ associative law. The order or position of the numbers has no influence on the outcome of addition and multiplication. The associative property can be used to address vector-related problems in vector algebra.