Equivalent (Cardinality) Sets Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

The notion of “equivalence” of sets’ cardinalities is a key to talking about infinite sets and comparing them to finite sets. This discussion helps you make sense of this idea.

Imagine that Spacey the space traveller is on an alien planet with a pile of diamonds and an alien. Spacey and the alien want to divide the pile of diamonds evenly between themselves (neither wants to trigger an interplanetary war by  seeming to take an unfair share), but they don’t speak the same language, and so in particular can’t name numbers in a way both will understand. Discuss how they might nonetheless divide the pile evenly (maybe with one diamond left over, if the total number is odd), and, more importantly, verify afterwards that the division was even.  What does this have to do with equivalence of sets?

Discuss how you could show that the set of perfect squares, {0,1,4,9,16,…}, has the same cardinality as the set of non-negative integers, {0,1,2,3,4,…}.