Question

question 7
is the sum of two irrational numbers closed under addition? justify your response.
no, the sum of any pair of opposite irrational numbers will be zero and zero is not an irrational number.
yes, the sum of any pair of opposite irrational numbers will be zero and zero is not an irrational number.
yes, the sum of any two elements within the set will be irrational and thus also in the set.
no, the sum of any two elements within the set will not be irrational and thus not in the set.
question 8
are the rational numbers closed under multiplication? justify your reasoning.
no, the product of any two rational numbers will also be a rational number.
yes, the product of any two rational numbers will also be a rational number.
yes, the product of any two rational numbers will not also be a rational number.
no, the product of any two rational numbers will not also be a rational number.

Explanation:

For Question 7: A set is closed under an operation if the result of the operation on any two elements in the set is also in the set. For irrational numbers, take opposites like $\sqrt{2}$ and $-\sqrt{2}$: their sum is $0$, which is rational (not irrational), so the set is not closed under addition.
For Question 8: A rational number can be written as $\frac{a}{b}$ where $a,b$ are integers and $b
eq0$. Multiplying two rationals $\frac{a}{b}$ and $\frac{c}{d}$ gives $\frac{ac}{bd}$, which is also rational, so rationals are closed under multiplication.

Answer:

Question 7: No, the sum of any pair of opposite irrational numbers will be zero and zero is not an irrational number.
Question 8: Yes, the product of any two rational numbers will also be a rational number.