Exercise 7:

Show that the function \(f : \mathbb{N} \to \mathbb{Z}\), defined by:

\[ f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even}, \\\\ -\frac{n+1}{2}, & \text{if } n \text{ is odd}. \end{cases} \]

is bijective.

Exercise 8:

Let \(A\) and \(B\) be two non-empty subsets of a set \(E\). Consider the function \(f : P(E) \to P(A) \times P(B)\), defined by:

\(f(X) = (X \cap A, X \cap B)\).

• Show that \(f\) is injective if and only if \(A \cup B = E\).
• Show that \(f\) is surjective if and only if \(A \cap B = \emptyset\).
• Give a necessary and sufficient condition on the sets \(A\) and \(B\) for \(f\) to be bijective.

Exercise 9:

Consider three sets \(A\), \(B\), \(C\), and two functions \(f : A \to B\), \(g : B \to C\). Show that:

• If \(g \circ f\) is injective, then \(f\) is injective.
• If \(g \circ f\) is surjective, then \(g\) is surjective.