91Ó°ÊÓ

If \(a \mid b\) and \(b \mid a\), show that \(a=\pm b\).

If the set \(S\) has a finite number of elements, prove: (a) If \(\sigma\) maps \(S\) onto \(S\) then \(\sigma\) is one-to-one. (b) If \(\sigma\) is a one-to-one mapping of \(S\) into itself, then \(\sigma\) is onto. (c) Prove, by example, that both part (a) and part (b) are false if \(S\) does not have a finite number of elements.

To check that \(n\) is a prime number, prove that it is sufficient to show that it is not divisible by any prime number \(p\), such that \(p \leq \sqrt{n}\).

Show that \(n>1\) is a prime number if and only if for any \(a\) either \((a, n)=1\) or \(n \mid a .\)

Prove that there is a one-to-one correspondence between the set of integers and the set of rational numbers.