Take two integers a and b. If you add a+ b you will find that it is equal to b+ a . i. e. Give the value of a as 1 and b as 2. when you add a + b i.e 1+ 2 you get 3. and if you add b+ a i. e. 2+1 again the answer is 3 itself. This is called commutative property of addition. The same holds good for multiplication as well.. i.e. if you multiply a x b you get the result same as when you multiply b with a. i.e. putting the values 1 and 2 to a and b correspondingly you get a xb = 1 x 2 = b x a = 2 x 1= 2 itself. This is called commutative property of multiplication. You can try any integer in place of a and b and will find that it stands for the same and will result another integer as the output.
To describe associative property you need to consider three intergers a, b and c. i. e. a + (b +c) = (a+b) ) + c i.e. to make it more clear we will take the value of a as 1 b as 2 and c as 3. , now 1+ (2 + 3) = (1 +2 ) +3
i.e. 1+ 5 =3+ 3.
6= 6. You see that both the values are same. This is called associative property of addition. The same holds good for multiplication also.
i.e a x (b xc) = (a x b) x c. Now put the values of a, b, and c as 1, 2 and 3.
i. e. 1 x (2 x3) = (1 x 2) x 3
i.e. 1 x 6 = 2 x 3
6 = 6. This proves that integers are closed under associative property of multiplication and we are getting an integer as result in all the cases.
By understanding the examples above you can see that the integers are closed under addition as well as multiplication in both commutative and associative properties.
