Chapter 3.4, Problem 31MS

Generalizing “equality.” Throughout this chapter we have considered the notion of same cardinality. If two collections have the same cardinality, then they are “equally numerous” (with the understanding that their sizes might be infinite). In this chapter we discovered that there are infinitely many different sizes of infinity. It turns out that “same cardinality” is a generalization of what we mean when we say two things are “equal”. In mathematics, we call such a generalization an equivalence relation. Specifically, an equivalence relation between two objects (say A and B) is any relation that is reflexive (that is, an object A must be equivalent to itself), symmetric (that is, if an object A is equivalent to an object B, then B must be equivalent to A), and transitive (that is, if an object A is equivalent to B and B, in turn, is equival ent to C, then A must be equivalent to C). Show that the relation of “same cardinality” is an equivalence relation between two collections.