Mathematics / foundation
Logic and the craft of proof
Develop precise arguments through quantified statements, proof techniques, induction, and constructive counterexamples.
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Inside this module
Module 01 foundation
Logic and the craft of proof
Develop precise arguments through quantified statements, proof techniques, induction, and constructive counterexamples.
- 01
Propositions and truth functions
Separate statements from open sentences and evaluate logical connectives without importing conversational assumptions.
22 min - 02
Quantified statements and their negations
Track domains and witness dependencies, then negate nested claims without losing their logical order.
22 min - 03
Direct proof and contraposition
Convert definitions into a chain of justified implications and choose a contrapositive when the negated conclusion is useful.
22 min - 04
Contradiction and existence proofs
Use an impossible consequence carefully and distinguish finding a witness from showing that a witness must exist.
22 min - 05
Mathematical induction and invariants
Connect a base case to a uniform successor step and recognize the same structure in maintained program properties.
22 min - 06
Strong induction and well-ordering
Use all smaller cases when a construction splits and connect induction to the impossibility of a smallest failure.
22 min - 07
Recursive definitions and structural induction
Define finite objects through constructors and prove a property by covering every way those objects can be built.
22 min - 08
Equivalence relations as proof tools
Use reflexivity, symmetry, and transitivity to group objects, then check that operations do not depend on a representative.
22 min - 09
Extremal and minimal counterexamples
Choose a measurable smallest failure or largest admissible object and extract information from the impossibility of improving it.
22 min - 10
Writing and checking a complete proof
Turn exploration into a readable argument with declared scope, justified dependencies, and adversarial checks at its weakest steps.
22 min