Question

question 5
closure means that if an operation is conducted on any two elements of the set,...
then the result of that operation is also within the set.
then the set has an infinite number of elements.
then the operation is commutative.
then the result is a new subset.

question 6
is the set of two rational numbers closed under addition? justify your reasoning.
no, the sum of any two elements within the set will not be a rational number and thus not in the set.
no, the sum of any two elements within the set will be a rational number and thus also in the set.
yes, the sum of any two elements within the set will be a rational number and thus also in the set.
yes, the sum of any two elements within the set will not be a rational number and thus not in the set.

Explanation:

Question 5

Closure is a fundamental property of sets under operations, defined as the result of the operation on any two set elements remaining in the set. The other options describe unrelated set/operation properties.

By definition, a rational number can be written as $\frac{a}{b}$ where $a,b$ are integers and $b
eq0$. Adding two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ gives $\frac{ad+bc}{bd}$, which is also a rational number (since $ad+bc$ and $bd$ are integers, $bd
eq0$). This satisfies closure under addition.

Answer:

then the result of that operation is also within the set.

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Question 6