Proofs

• Can We Prove the Irrationals Are Uncountable Without Cantor’s Diagonal Argument?
  Yes. A nested-interval argument proves the real numbers, and therefore the irrationals, are uncountable without decimal diagonalization.
• How to Prove a Relation Is Reflexive and Transitive
  Learn the definitions and proof pattern for reflexivity and transitivity, with worked examples and common mistakes to avoid.
• Examples of Open Problems Solved by Short Proofs
  See examples where a compact argument settled a mathematical problem, and learn why a short proof can hide a long search or earlier work.
• Which Discrete Mathematics Books Are Best for Computer Science Self-Study?
  For computer science self-study, compare Rosen, Epp, and MIT's free Mathematics for Computer Science, then choose one book and work its exercises.
• How Does a Chain of Implications Prove Conditions Equivalent?
  A chain proves several conditions equivalent when its implications connect every condition and eventually return to the starting one.
• 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.