Discrete mathematics

• Prove That If x + 5 Is Odd, Then x² Is Even
  For integer x, x+5 odd forces x even, so x² is even. See the direct proof, a numerical example, and why the integer assumption matters.
• Is (A ∩ B) ∪ C = A ∩ (B ∪ C)?
  The proposed identity is false for arbitrary sets. See a counterexample, the correct distributive law, and the exact condition for equality.
• What Is the Truth Value of P(x,y,z)?
  The predicate P(x,y,z) is true exactly when x+y is at most z, but the missing expression is needed to determine the requested truth value.
• How Do You List the Elements of a Set?
  A set’s elements can be listed only when its definition is known. Learn how to expand set-builder notation and write finite or infinite sets.
• 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.
• Why You Cannot Connect 3 Dots to 3 Squares Without Crossings
  The three-dot, three-square puzzle requires drawing K3,3. Euler’s formula shows it needs nine edges, but a planar bipartite graph on six vertices allows at most eight.
• How to Graph a Relation and Identify Its Properties
  Learn how to draw a relation as a directed graph and test reflexivity, symmetry, antisymmetry, and transitivity with a worked finite example.
• 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.
• What Is a Counterexample to ∀x∃y (x = 1/y)?
  Over the real numbers, x = 0 is a counterexample: no nonzero real y has reciprocal zero. The answer depends on the domains.
• How Many Ways Can You Climb n Stairs Using 2 or 3 at a Time?
  Count ways to climb n stairs using only 2- or 3-stair moves: derive the recurrence, generating function, and an explicit binomial-sum formula.