Q: Are the sets of positive fractions closed for addition?

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Yes.

No. 1 + 3 = 4, which is not odd. In fact, no pair of odds sums to an odd. So the set is not closed under addition.

Closed sets and open sets, or finite and infinite sets.

Closed sets and open sets, or finite and infinite sets.

They are equivalent fractions as for example: 3/4 = 9/12

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Yes.

I know that whole numbers, integers, negative numbers, positive numbers, and even numbers are. Anyone feel free to correct me.

No. 1 + 3 = 4, which is not odd. In fact, no pair of odds sums to an odd. So the set is not closed under addition.

Closed sets and open sets, or finite and infinite sets.

Closed sets and open sets, or finite and infinite sets.

Closed sets and open sets, or finite and infinite sets.

No, it is not.

You cannot: whole numbers and improper fractions are disjoint sets.

yes, since every closed set can be written as a intersection of open sets. (Recall that borel sets is sigma algebra)

Closed sets and open sets, or finite and infinite sets.

They are equivalent fractions as for example: 3/4 = 9/12

To be closed under an operation, when that operation is applied to two member of a set then the result must also be a member of the set. Thus the sets ℂ (Complex numbers), ℝ (Real Numbers), ℚ (Rational Numbers) and ℤ (integers) are closed under subtraction. ℤ+ (the positive integers), ℤ- (the negative integers) and ℕ (the natural numbers) are not closed under subtraction as subtraction can lead to a result which is not a member of the set.