Commutative And Associative
Definition of commutative and associative rules:
Commutative: General meaning of commutative is changing the order of the operands does not change the end result in a binary operation.
Associative: Means that when one adds more than two numbers, order in which addition is performed does not matter.
Commutative and Associative are the basic and fundamental properties of mathematics. Every branch of mathematics satisfies these fundamental laws. Apart from these two, there is one more law known as Distributive law.
Explanation of Commutative and Associative Rules:
Let us consider that you have ten apples and two bags in your hands. First put 3 apples into the first bag. So first bag contain 3 apples at present. Now put the remaining 2 apples also. Totally there are 5 apples in the bag now.
Now take the second bag and put first 2 apples of remaining 5 apples in it. So second bag has 2 apples now. Now put the remaining 3 apples also in it. Now the second bag also contains 5 apples as in the first bag.
Conclusion: The way we put the apples in the bag doesnt alter the result. This is the basic aim of commutative law.
i.e., reversing the operands in a binary sum from left to right and right to left yield the same result.
For example, if a and b are any two numbers, then
a + b = b + a (Known as Commutative Law of Addition).
i.e., specifically 4 + 5 = 9 & 5 +4 = 9.
This holds good for multiplication also.
i.e., a * b = b * a (Called as Commutative law of Multiplication).
Note: Commutative law holds good for binary operations such as Addition and multiplication only. Other binary operations, subtraction and division are not commutative.
Associative property: As stated earlier Associative means, when one adds more than two numbers, order in which addition is performed does not matter.
Let us consider the following example to get a clear idea. Here there are three different colored coins available.3 blue,1 green and 2 black colored coins.
The total no of coins present can be obtained by adding in either of the following way.
2+(1+3)
Or (2+1) +3.
Both yield the same result. i.e., 6 coins.
Thus in multiple additions the order considered for addition doesnt matter.
In general
(a + b) + c = a + (b + c).
Is known as Associative property.
Note: Multiplication also hold good for associative law. i.e.,
(a * b) * c = a * (b * c).
Problems Related to Commutative and Associative Rules:
1) Prove the equality 3 + 4 = 4 + 3 by commutative property.
Sol:
Take Left hand side of equality. i.e., 3 + 4 = 7 ________(1)
Similarly Right hand side is
4 + 3 = 7__________(2)
Here, Both equations(1)&(2) yields same result. So the given equality is true.
Practice Problem:
1) Prove the equality (1 + 2) + 3 = 1+ (2 + 3) by associative property.
2) Prove the equality (1 * 2) * 3 = 1* (2 * 3) by associative property.
by: nayaknandan
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