Euler's totient via inclusion-exclusion

Sets. For each prime $p\mid n$, $A_p=\{$multiples of $p$ in $\{1,\ldots,n\}\}$; $|A_p|=n/p$.

IE. Number coprime to $n$ = $n$ minus those in the union:
$$\varphi(n)=n-\sum_p\frac np+\sum_{p<q}\frac{n}{pq}-\cdots=n\prod_{p\mid n}\left(1-\frac1p\right).$$

Conclusion. $\varphi(n)=n\prod_{p\mid n}(1-1/p)$.