True or false? Negative numbers are closed under addition. Give a counterexample

True or false? Prime numbers are closed under subtraction. Give a counterexample
True or false? Natural numbers are closed under division. Give a counterexample

I'll do one; you try the others.

True or false? Negative numbers are closed under addition.

True: for any positive a,b
-a + -b = -(a+b)

False

False. Negative numbers are not closed under addition.

Counterexample: Let's consider the numbers -2 and -3. When we add them, -2 + (-3) = -5, which is also a negative number. However, -5 is not a negative number as we started with.

False. Prime numbers are not closed under subtraction.
Counterexample: Let's consider the prime numbers 5 and 7. When we subtract them, 5 - 7 = -2, which is not a prime number.

True. Natural numbers are closed under division.
Counterexample: If we take any two natural numbers, for example, 6 and 3, and divide them, we get 6 ÷ 3 = 2, which is also a natural number. Hence, they are closed under division.

False. Negative numbers are not closed under addition. To confirm this, we can provide a counterexample. For instance, if we add -5 and -3, the result is -8, which is also a negative number. However, -8 is not a natural number, so negative numbers are not closed under addition.

True. Prime numbers are closed under subtraction. To understand this, we need to consider that any two prime numbers can be subtracted from each other to yield an integer. For example, if we subtract 7 from 13, we get 6, which is an integer. Therefore, prime numbers are closed under subtraction.

False. Natural numbers are not closed under division. We can demonstrate this with a counterexample. If we divide 6 by 4, the result is 1.5, which is not a natural number because natural numbers are whole numbers without fractions or decimals. Thus, natural numbers are not closed under division.