What is associative property and commutative property?

Answer 1

If a binary operation is associative, it means that you get the same result if you change the order. For example, let * denote a binary operation. Then, if * is associative,

a*(b*c) = (b*c)*a

This would hold if, for example, * represents the operation of addition. It would not hold if * represents subtraction.

eg 1+(1+2) = (1+2)+1 = 4

but 1-(1-2) = 2, whereas (1-2)-1 = -2

If a binary operation is commutative, then you get the same result no matter what order you do the operations in. So,

a*(b*c) = (a*b)*c

Here, you can see that multiplication is commutative, but division is not.

e.g. 1x(1x2) = (1x1)x2 = 2

but 1/(1/2) = 2, whereas (1/1)/2 = 1/2

* * * * *

The above answer is completely the wrong way around.

Associativity implies that

(a * b) * c = a * (b * c) and so either can be written as a * b * c

Commmutativity implies that

a * b = b * a

Multiplication is associative as well as commutative whereas division is neither.