Proof foundations: from claims to arguments
A twelve-lesson proof foundations unit shared by analysis, algebra and discrete mathematics. Exact checks support practice; authored arguments require review. This unit is not the complete undergraduate discrete mathematics course.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
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Proof foundations: argument portfolio assessment
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Proof portfolio: from a remainder procedure to a theorem
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
A claim needs a domain
Lesson details and objectives
Separate examples, universal claims and counterexamples by identifying the allowed objects.
- State a domain and a predicate precisely.
- Verify a counterexample against every assumption.
4 practice tasks.
Who chooses first?
Lesson details and objectives
Track how witnesses depend on universally chosen values.
- Distinguish forall-exists from exists-forall.
- Construct a witness and refute a proposed uniform witness.
6 practice tasks.
Negate the whole claim
Lesson details and objectives
Turn failed universal claims into precise witnesses without overclaiming.
- Negate nested quantifiers.
- Separate some missing work from no work by a learner.
4 practice tasks.
Unpack definitions to build a proof
Lesson details and objectives
Use integer witnesses rather than treating divisibility as unexplained symbolism.
- Translate divisibility into an integer equation.
- Build a direct proof using closure under integer arithmetic.
4 practice tasks.
Reverse and negate with care
Lesson details and objectives
Prove an implication through its contrapositive while keeping the converse separate.
- Form the contrapositive correctly.
- Prove a parity implication with integer witnesses.
4 practice tasks.
Contradict the exact assumption
Lesson details and objectives
Use a constructed smaller object to refute existence of a least positive rational.
- Identify the assumption to negate.
- Verify a constructed object satisfies every required condition.
4 practice tasks.
Cover every case and both directions
Lesson details and objectives
Organize parity arguments and understand what an if-and-only-if claim requires.
- Use exhaustive, domain-appropriate cases.
- Prove each direction of a biconditional.
4 practice tasks.
Prove two sets contain the same objects
Lesson details and objectives
Translate membership into logic and use double inclusion.
- Distinguish membership from subset relations.
- Prove a set identity and diagnose a false variation.
4 practice tasks.
Follow membership through a function
Lesson details and objectives
Compare inverse images, which preserve intersections, with images, which can merge distinct inputs.
- Use domain, codomain, image and inverse-image definitions.
- Prove an inverse-image identity without assuming an inverse function exists.
4 practice tasks.
Which properties survive composition?
Lesson details and objectives
Trace witnesses through two functions and distinguish properties of the composite from those of its factors.
- Apply injectivity and surjectivity definitions.
- Prove a composition implication and construct a counterexample to a different one.
6 practice tasks.
Make the next case depend on proved cases
Lesson details and objectives
Distinguish a tested pattern from an induction argument with adequate base cases.
- Prove a summation by induction.
- Use strong induction with enough base cases for a three-step recurrence.
4 practice tasks.
Group objects by a proved equivalence
Lesson details and objectives
Establish an equivalence relation, build its classes and assemble a reviewable proof portfolio.
- Prove reflexivity, symmetry and transitivity separately.
- Distinguish finite checks, general proof and reviewed evidence.
4 practice tasks.