Question

a third grader asked his teacher whether infinite sets could be compared. how might you respond? choose the correct answer below. a. yes. infinite sets can be compared. two infinite sets are equal as long as it is possible to find a one - to - one correspondence between them. b. no. infinite sets cannot be compared. infinite sets have infinite cardinality and the concept of one - to - one correspondence cannot be applied to them. c. no. infinite sets cannot be compared. all infinite sets are subsets of all other infinite sets, so the comparisons are not meaningful. d. yes. infinite sets can be compared. as long as it is possible to find a one - to - one correspondence between two infinite sets, they are equivalent.

Explanation:

To determine if infinite sets can be compared, we use the concept of one - to - one correspondence. If a one - to - one correspondence can be established between two infinite sets, they can be compared (e.g., the set of natural numbers and the set of even numbers can be put in one - to - one correspondence). Option A is wrong as it says two infinite sets are equal if there's a one - to - one correspondence (they are equinumerous, not necessarily equal). Option B is wrong as one - to - one correspondence can be applied to infinite sets. Option C is wrong as infinite sets are not all subsets of each other (e.g., the set of real numbers and the set of natural numbers) and comparisons are meaningful. Option D correctly states that infinite sets can be compared via one - to - one correspondence.

Answer:

D. Yes. Infinite sets can be compared. As long as it is possible to find a one - to - one correspondence between two infinite sets, they are equinumerous.