MATH1961 · Mathematics 1a (advanced)
Mathematical Foundations
This is the chapter that makes MATH1961 Advanced. Before any calculus, you build the rigour toolkit the whole unit runs on: how to read a statement exactly (the quantifier order is part of the answer), the four standard proof techniques, mathematical and strong induction, and the property that separates ℝ from ℚ — completeness, the Least Upper Bound Axiom. The lecturer grades logic: a misplaced quantifier or a confused implication arrow loses marks even when the underlying idea is right, and a proof is expected to be “a mixture of English and mathematical symbols, written in full sentences with correct grammar.” You must negate quantified statements (one well-chosen counterexample is a full proof), pick the right machine — direct, contrapositive, contradiction, or cases — run an induction cleanly, and state what a supremum is, in both clauses.
What this chapter covers
- 011.1 Notation you must use correctly — quantifiers, implication, the number-system chain
- 021.2 Negating quantified statements — the counterexample as a complete proof
- 031.3 The four proof techniques: direct, contrapositive, contradiction, cases
- 04Mathematical and strong induction — the worked template
- 05The completeness of ℝ — sup / inf and the Least Upper Bound Axiom
Worked example: prove sup of a bounded set, both clauses
- +1State what sup A = 1 means. You must show (i) 1 is an upper bound of A, and (ii) for every ε > 0, 1 − ε is not an upper bound — the least-upper-bound conditions.
- +1(i) Upper bound. For every n ∈ ℕ, 1/n > 0, so 1 − 1/n < 1. Hence every element of A is ≤ 1, and 1 is an upper bound.
- +1(ii) Nothing smaller works. Take any ε > 0. By the Archimedean property choose n with 1/n < ε, i.e. n > 1/ε. Then 1 − 1/n > 1 − ε.
- +1Conclude (ii). So 1 − ε is exceeded by an element of A, meaning 1 − ε is not an upper bound for any ε > 0.
- +1Combine. 1 is an upper bound and no smaller number is, so 1 is the least upper bound: sup A = 1. (Note 1 ∉ A, so the sup need not be attained.) ■
Key terms
- Quantifier order
- Whether a statement reads ‘for all x there exists y’ or ‘there exists y for all x’ — the two mean different things. In the ε–δ definition the order for-all-ε then there-exists-δ is the whole point: δ is allowed to depend on ε. Swap them and you have written something false, and you lose the marks that live in that order.
- Counterexample
- A single instance that makes a ‘for all’ claim false. To disprove a universal statement you prove its negation, and exhibiting one well-chosen counterexample is a complete proof — it earns full marks. Classic traps: ‘bounded implies integrable’ (false, by the Dirichlet function) and ‘critical point implies extremum’ (false, x³ at 0).
- Contrapositive
- The statement ‘not Q implies not P’, which is logically equivalent to ‘P implies Q’. When a direct proof stalls, proving the contrapositive is often clean — e.g. ‘if n² is even then n is even’ is awkward directly but immediate from its contrapositive (assume n odd, square it).
- Mathematical induction
- A proof method for statements indexed by ℕ: prove the base case P(1), then prove the inductive step P(k) implies P(k+1); together they give P(n) for all n. Strong induction assumes P(1),…,P(k) all hold to prove P(k+1), which is needed when a step depends on more than the immediately preceding case.
- Supremum (least upper bound)
- The smallest number that is ≥ every element of a set A. It satisfies two clauses: it is an upper bound, and no smaller number is. The completeness (Least Upper Bound) Axiom guarantees every non-empty set bounded above has a supremum in ℝ — the property ℚ lacks — and it underwrites the IVT, EVT and monotone convergence.
Mathematical Foundations FAQ
Why does MATH1961 spend a whole chapter on logic and proof before any calculus?
Because the Advanced unit grades whether you can construct an argument, and the same machinery — quantifiers, the proof methods, induction, and the completeness of ℝ — is what later proves the IVT, the EVT, the MVT and the FTC. Master the spine first and the rest of the course becomes ‘apply the toolkit’. The quantifier discipline you build here is exactly what an ε–δ limit proof tests.
How do I know which proof technique to use?
Read the statement first. A ‘P implies Q’ you can attack head-on takes a direct proof; if direct stalls (especially with ‘even/odd’ or divisibility), try the contrapositive; for ‘there is no…’ or irrationality claims use contradiction; when the hypothesis splits into mutually exclusive situations use cases; and for a statement indexed by ℕ, use induction. Naming the technique is itself part of a clean answer.
What exactly do I have to write for a supremum proof?
Two clauses, always. First show the candidate value is an upper bound (every element is ≤ it). Then show no smaller number is an upper bound: take any ε > 0 and produce an element of the set exceeding (candidate − ε), usually via the Archimedean property. Both clauses are required — omitting the second is the most common way to lose half the marks.
Is a single counterexample really enough to disprove a statement?
Yes — for a ‘for all’ claim, one counterexample is a full proof of its negation, and it earns full marks. The skill is choosing a clean one. For a ‘there exists’ claim it is the opposite: one witness proves it true. Match the witness to the quantifier.
Exam move
Drill the rigour toolkit until it is reflex, because every later proof reuses it. Make flashcards of the notation and the negation rules (push the negation inward, flip each quantifier). For each of the four proof methods keep one worked template you can reproduce on a blank page. Practise supremum proofs as a fixed two-clause ritual (upper bound, then nothing smaller via Archimedes). For induction, write the base case, the explicit inductive hypothesis, and the step as three labelled lines — markers reward the structure. Above all, write in full sentences mixing English and symbols, with the quantifiers in the right order: that style is what the lecturer grades.
