Let \(Z\) be a set and let \(X_1, X_2, \dots\) be subsets of \(Z\). Prove the formula \(\bigl(\bigcap_i X_i\bigr)^c = \bigcup_i X_i^c\), where \((\cdot)^c\) denotes complement.
Solution
For any \(z\in Z\), \[ \begin{aligned} z\in\left(\bigcap_iX_i\right)^c &\Longleftrightarrow z\notin\bigcap_iX_i\\ &\Longleftrightarrow \text{there exists }i\text{ such that }z\notin X_i\\ &\Longleftrightarrow \text{there exists }i\text{ such that }z\in X_i^c\\ &\Longleftrightarrow z\in\bigcup_iX_i^c. \end{aligned} \] Since the two sets have exactly the same elements, \[ \left(\bigcap_iX_i\right)^c=\bigcup_iX_i^c. \]