Question

☺Hey peeps!☺

⭐Are natural numbers closure under subtraction?

⭐Are whole numbers distributive under subtraction?

⭐Are integers associative under addition?

⭐Are natural numbers distributive under subtraction?

Thanks!

@$∆M

Answer 1

answer 1):-The integers (symbol ) are the set of all of the natural numbers, plus their additive inverses (their negatives), and zero {...-4, -3, -2, -1, 0, 1, 2, 3, 4, ....} Theintegers are "closed" under addition,multiplication and subtraction, ... Therational numbers are "closed" underaddition, subtraction, andmultiplication.

answer2):-Distributive Property of Multiplicationover Subtraction. This property tells that multiplication of a whole numberis distributed over the difference of thewhole numbers. Suppose 'a', 'b', and 'c' are three whole numbers.



answer 3):-Associative Property of Addition.Adding integers will have the same result regardless of the grouping. The sum will not change even if theintegers are grouped differently.


answer 4):-12 is a natural/whole number. Therefore, the system is closed underaddition. 2) Subtraction: Subtractionof two whole or natural numbers does not always result in a whole number ornatural number. For example, take any two natural numbers, say 3 and 9.

Answer 2

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1. No

1. Subtraction of two whole or natural numbers does not always result in a whole number or natural number. For example, take any two natural numbers, say 3 and 9. Now, 3 – 9 = -6. -6 is not a natural/whole number. Therefore, the system is not closed under addition.

2. Yes

2. Multiplication of whole numbers is distributive over addition.

If a, b and c are any three whole numbers,

then a x (b + c) = ab + ac

Example :

2 x (3 + 4) = 2x3 + 2x4 = 6 + 8 = 14

2 x (3 + 4) = 2x (7) = 14

Hence, 2 x (3 + 4) = 2x3 + 2x4

Therefore, Multiplication is distributive over addition.

3. Yes

3. Observe the following example :

Consider the integers 5, – 4 and 7.

Look at

5 + [(– 4) + 7] = 5 + 3 = 8

and

[5 + (– 4)] + 7 = 1 + 7 = 8

Therefore, 5 + [(– 4) + 7] = [5 + (– 4)] + 7

In general, for any integers a, b and c, we can say,

a + (b + c) = (a + b) + c.

Therefore addition of integers is associative.

4. Yes

4. Multiplication of natural numbers is distributive over subtraction.

If a, b and c are any three natural numbers,

then a x (b - c) = ab - ac

Example :

2 x (4 - 1) = 2x4 - 2x1 = 8 - 2 = 6

2 x (4 - 1) = 2 x (3) = 6

Hence, 2 x (4 - 1) = 2x4 - 2x1

Therefore, multiplication is distributive over subtraction.

THANKS FOR ASKING SUCH A QUESTION DEAR FRIEND ☺️✌️