Proofs

• How to Prove the Fundamental Hyperbolic Identity
  Prove cosh²(x) − sinh²(x) = 1 directly from exponential definitions, then use it to derive related hyperbolic identities and check examples.
• Why Does a Tree with n Vertices Have n − 1 Edges?
  Prove that every finite tree with n vertices has n − 1 edges by induction, using the fact that each nontrivial tree has a leaf.
• Show That the Identity Function on Any Set Is a Bijection
  For any set A, the identity map sends each element to itself, making it injective and surjective—and therefore bijective, even when A is empty.