Definition: one-to-one and onto functions

A function \(f: A \rightarrow B\) is one-to-one (1-1) if \(a_1 \neq a_2\) in \(A\) implies that \(f(a_1) \neq f(a_2)\) in \(B\). The function \(f\) is onto if, given any \(b \in B\), it is possible to find an element \(a \in A\) for which \(f(a) = b\).


Definition: Correspondance

The set \(A\) has the same cardinality as \(B\) if there exists \(f: A \rightarrow B\) that is 1-1 and onto. In this case, we write \(A \thicksim B\).


This can also be written as \(\displaystyle \lim_{n\to\infty}a_n = a\). What is this saying? After a certain number of terms, say after \(N\), all the points in the sequence will converge to a real number \(a\). Defining these points with \(|a_n - a| \leq \epsilon\) is explained in the next definition.

Definition: Countable Sets

A set \(A\) is countable if \(N \thicksim A\). An infinite set that is not countable is called an uncountable set.



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