Question

what is closure property, commutative property ,associative property and distributive property of addition and multiplication .

Answer 1

CLOSURE PROPERTY OF ADDITION., a+b=c, eg 2+3=5....
MULTIPLICATION =a*b=c, eg =2*3=6...
COMMUTATIVE PROPERTY OF ADDITION=a+b=b+a, eg, 2+3=3+2...
MULTIPLICATION, a*b=b*a, eg, 2*3=3*2...
ASSOCIATIVE PROPERTY OF ADDITION, a+(b+c)=(a+b) +c....
MULTIPLICATION,a*(b*c) =(a*b) *c...
DISTRIBUTIVE PROPERTY OF ADDITION, (a+b) =c=(b+a)...
MULTIPLICATION, (a*b) =c=(b*a) …..
Hope this will help you. ..

Answer 2

1-Closer property

If x and y are any two natural number then if you add them together you will get a a result in Natural number too..for example 2 and 5 are natural number , if we add them together we get result 2 + 3 = 5 , 5 is a natural number too...This is a one of the property of natural number so called Closer property...
Here we use 'Addition' operation between the number So we call this property as Closer property under Addition...

Similarly if we multiply any two natural number again we get a result in Natural number too...For example- 3 x 2 = 6 , here 2 and 3 are natural numbers and their results 6 again a natural number...This is also a closer property of number...As we use Multiplication operation here So called Closer property under Multiplication...

This property is not valid only for two natural numbers..but it is also true for three or more natural numbers...

Definition of Closer property

Closer property under Addition :

If x and y are any two natural numbers then x + y is also a natural number...

Closer Property under Multiplication:
If x and y are any two natural numbers then x X y is also a natural number...

Note : Closer property is valid for natural number under Addition and Multiplication...

This property fails for Subtraction and Division..For example 3 and 5 are natural numbers but 3 - 5 = -2 , here -2 is not a natural number. Thus closer property under subtraction fails for natural number...
Simillarly 3 and 5 are natural numbers but 3/5 is not a natural number....Closer property is also not valid for division...

Closer Property also holds for whole Numbers , Integers , Rational numbers , Real numbers under Addition and Multiplication...You can choose any two nimbers from the mentioned Number system you will get the result of the same number system...

Closer Property is true for Integers , Rationals and Real numbers under subtraction...
For example -> 2 and 6 are integers , 2 - 6 = -4
Here -4 is also an integer...Similary you can check for other number system....

2 -Commutative Property

Commutative Property under Addition
For any two natural number a and b
a + b = b + a
This property is known as Commutative property under Addition(See the opertaion between the number) For example : 2 +3 = 3 +2


Commutative Property under Multiplication:
a x b = b x a where a and b are any natural numbers...
For example : 2 × 3 = 3 × 2

Note : Commutative property is true(Valid) for Natural , Whole , Integers , Rational , Irrationals and Reals numbers under Addition and Multiplication...

But Commutative fails for subtration and Division..
Example : 3 - 2 = 1 but 2 - 3 = -1
here 3- 2 is not equal to 2 - 3..so commutative property under subtraction fails...

Similary, For Division... 4/2 = 2 but 2/4 = 1/2
Clearly , 4/2 is not equal to 2/4...So Commutative property is not true for division..

3 - Associative Property :
1- a + ( b + c ) = ( a + b ) + c under Addition
2- a x ( b x c ) = (a x b ) x c. under Multiplication

For example :
Under Addition :
2 + ( 3 + 4 ) = 2 + 7 = 9
( 2 + 3 ) + 4 = 5 + 4 = 9
Thus LHS = RHS...

Under Multiplication :
2 × ( 3 × 4 ) = 2 × 12 = 24
( 2 × 3 ) × 4 = 6 × 4 = 24..
LHS = RHS..
Again this property fails for subtraction and division..

Under subtarction :
2 - ( 3 - 4 ) = 2 - ( - 1 ) = 2 + 1 = 3
( 2 - 3 ) - 4 = -1 - 4 = - 5
Here LHS is not equal to RHS...So Associative Property under Subtraction fails.

you can also check this property for division by taking any three number and you wil find that the property again fails...

4 - Distributive Property
a x ( b + c ) = a x b + a x c

Example :

2 × ( 4 + 2 ) = 2 × 6 = 12
2 × 4 + 2 × 2 = 8 + 4 = 12
Thus LHS = RHS...

Hope this help....