Nature of proofs

This syllabus develops students’ ability to understand and construct mathematical proofs, building a strong foundation in logical reasoning and rigorous argument. Students learn the formal language and notation used in mathematics, including statements, implications, converses, negations, equivalence, and contrapositives, and how these are used to form valid logical arguments. They explore different proof techniques such as proof by contradiction, counterexamples, and mathematical induction, and apply these methods to prove results involving integers, inequalities, algebraic identities, sequences, and calculus. The course also develops deeper insight into inequalities (including the triangle inequality and arithmetic–geometric mean), recursive formulas, and the logical structure behind mathematical results, helping students communicate mathematical reasoning clearly and solve complex problems with precision.