TABLE OF CONTENTS In this chapter we introduce the concept of equivalence for sets and study countable sets. We also discuss briefly the axiom of choice.
Two finite sets A and B have the same number of elements if there exists a bijection of A onto B. We can count the visitors in a sold-out theatre by counting the seats. The concept of equivalence for sets is an extension of the concept of two sets having the same number of elements, generalised to infinite sets.
A set A is said to be equivalent to a set B if there exists a bijection of A onto B; we then write A~ B.
The set 
and the set of even positive integers are equivalent. Indeed, n 
2 n is a bijection of onto {2, 4, 6, }.
~ . Let f: n 2 n for n>0, and f :n ?2 n+1 for n ?0. Then f is obviously onto and it is easy to check that it is one-to-one. Hence it is a bijection of onto .
]0, 1[~]0, ?[. The required bijection is 
.
We observe that the relation A~ B is reflexive, symmetric and transitive, and thus it is an equivalence relation. Reflexivity follows from the use of the function id A, which provides a bijection of A onto...
