Natural numbers are closed under division: false.
A set is closed under a certain operation if the results of that operation are always inside that set.
So, if natural numbers were closed under division, the division of two natural numbers would always be a natural number.
You have plenty of counterexamples, to pick one you may divide any odd number by 2: 5/2 is not a natural number.
Negative numbers are closed under addition: true.
Let [tex]m,n[/tex] be two positive numbers. So, [tex]-m,-n[/tex] are two negative numbers. Their sum is
[tex](-m)+(-n)=-(m+n)[/tex]
And since [tex]m+n[/tex] is positive, we deduce that [tex]-(m+n)[/tex] is negative, so the sum of two negative numbers is still negative.
Prime numbers are closed under subtraction: false.
This would mean that the subtraction of two primes is also a prime. Again, there are many counterexamples: 7 is prime and so is 3, but their difference 7-3 is 4, which is not prime.
16.Natural number are closed under division: False
The set of natural number are closed under addition and Multiplication.
The set of natural numbers are not closed under subtracting and division.
Addition:
When [tex]a[/tex] and [tex]b[/tex] are the two natural numbers, [tex]a+b[/tex] are also natural number.
For example,
[tex]4+5=9[/tex]
[tex]9[/tex] is also a natural numbers.
Multiplication:
When [tex]a[/tex] and [tex]b[/tex] are the two natural numbers, [tex]a*b[/tex] are also natural number.
For example,
[tex]6*5=30[/tex]
[tex]30[/tex] is also natural numbers.
Division:
When [tex]a[/tex] and [tex]b[/tex] are the two natural numbers, [tex]a\div b[/tex] are not a natural number.
For example,
[tex]5\div3=1.4[/tex]
[tex]1.4[/tex] is a rational number not a natural number.
Thus, the natural number are not closed under division.
17.Negative number are closed under addition : True
The negative number is closed under addition.
Let [tex]a,b[/tex] be the two positive number and [tex]-a, -b[/tex] is two negative number.
Adding the two negative number is also a negative number.
Now,
[tex](-a)+(-b)=-(a+b)[/tex]
Thus, the negative number is closed under addition.
18.Prime numbers are closed under subtraction : False
Given, subtracting the two prime is also prime, but subtracting the two prime is not a prime ,except [tex]2[/tex] .
For example:
[tex]7[/tex] is prime and [tex]11[/tex] is prime now their difference is,[tex]7-11=4[/tex]
Which is not prime.
Hence, prime number are not closed under Subtraction.
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