Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

All ODE PDE abelian analyticity anova baire calculus of variations canonical forms cauchy-riemann compactness completeness complex conformal connectedness continuity contour integration convergence differentiation distributions fields fixed point galois green groups harmonic injectivity integral equations integration laurent limsup linear algebra liouville lp markov measure mechanics metric spaces multivariable multivariate numerical p-groups permutations polynomials power series probability quadratic forms real analysis regression residues rings sampling separability separation sequences series singularities stability statistics sylow testing topology uniform convergence wronskian zeros

#1 · Analysis & Linear Algebra › Metric Spaces

“Closed and bounded ⇒ compact” — false

Counterexample: Closed unit ball in ℓ² (or in C[0,1] with sup norm)

$e_{1}, e_{2}, \dots$ has no convergent subsequence since ‖ $e_{n} - e_{m}$ ‖ $= 2$ . Heine–Borel is $R^{n} -$ only.

compactnessmetric spaces

#2 · Analysis & Linear Algebra › Metric Spaces

“Bounded ⇒ totally bounded” — false

Counterexample: ℝ with the discrete metric

Everything is within distance 1, but no finite set of balls of radius ½ covers it.

compactnessmetric spaces

#3 · Analysis & Linear Algebra › Metric Spaces

“Completeness is a topological property” — false

Counterexample: (0,1) and ℝ

Homeomorphic, but $R$ is complete and (0,1) is not (1/n is Cauchy).

completenessmetric spaces

#4 · Analysis & Linear Algebra › Metric Spaces

“Connected ⇒ path-connected” — false

Counterexample: Topologist's sine curve {(x, sin 1/x) : 0 < x ≤ 1} ∪ {0}×[−1,1]

Connected as the closure of a connected set; no path reaches the segment.

connectednesstopology

#5 · Analysis & Linear Algebra › Continuity and Differentiation

“Continuous on a bounded interval ⇒ bounded” — false

Counterexample: f(x) = 1/x on (0,1)

Needs a compact (closed) domain.

continuity

#6 · Analysis & Linear Algebra › Continuity and Differentiation

“Uniformly continuous ⇒ Lipschitz” — false

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continuity

#7 · Analysis & Linear Algebra › Continuity and Differentiation

“Differentiable ⇒ continuously differentiable” — false

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differentiation

#8 · Analysis & Linear Algebra › Sequences and Series of Functions

“Pointwise limit of continuous functions is continuous” — false

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uniform convergence

#9 · Analysis & Linear Algebra › The Real Line

“Σaₙ convergent ⇒ Σaₙ² convergent” — false

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series

#10 · Analysis & Linear Algebra › The Real Line

“aₙ → 0 ⇒ Σaₙ converges” — false

Counterexample: Harmonic series Σ1/n

series

#11 · Analysis & Linear Algebra › Integration

“Riemann integrable ⇒ continuous almost everywhere fails for… (i.e. 'bounded ⇒ integrable')” — false

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integration

#12 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“A subset of ℝ with measure zero is countable” — false

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measure

#13 · Complex Analysis, Algebra & Topology › Topology

“Every subspace of a separable metric space is separable — fails for general topological spaces” — false

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topologyseparability

#14 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic polynomial ⇒ similar” — false

Counterexample: 0 matrix and [[0,1],[0,0]]

Both have char poly $x^{2}$ , different minimal polynomials (x vs $x^{2})$ .

linear algebracanonical forms

#15 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Diagonalisable ⇒ invertible” — false

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linear algebra

#16 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Real matrix with real eigenvalues is diagonalisable” — false

Counterexample: [[1,1],[0,1]]

Eigenvalue 1 with geometric multiplicity 1 < algebraic 2.

linear algebra

#17 · Analysis & Linear Algebra › Determinants and Matrix Tricks

“AB = I ⇒ BA = I for all matrices” — false

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linear algebra

#18 · Analysis & Linear Algebra › Inner Product Spaces and Forms

“Every real symmetric matrix is positive definite if det > 0” — false

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linear algebraquadratic forms

#19 · Complex Analysis, Algebra & Topology › Groups

“Converse of Lagrange: d | |G| ⇒ subgroup of order d” — false

Counterexample: A₄ has no subgroup of order 6

groups

#20 · Complex Analysis, Algebra & Topology › Groups

“H ⊴ K and K ⊴ G ⇒ H ⊴ G” — false

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groups

#21 · Complex Analysis, Algebra & Topology › Groups

“Every group of order p² is cyclic” — false

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groupsp-groups

#22 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every integral domain is a UFD” — false

Counterexample: ℤ[√−5]: 6 = 2·3 = (1+√−5)(1−√−5)

rings

#23 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every UFD is a PID” — false

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rings

#24 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every PID is Euclidean” — false

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rings

#25 · Complex Analysis, Algebra & Topology › Rings and Fields

“Irreducible over ℤ ⇒ irreducible mod every prime” — false

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polynomialsfields

#26 · Complex Analysis, Algebra & Topology › Cauchy Theory

“Bounded real part ⇒ entire function is constant — fails if only |f| is bounded on a half-plane” — false

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complexliouville

#27 · Complex Analysis, Algebra & Topology › Singularities and Residues

“Zeros of a non-constant analytic function can accumulate inside the domain” — false

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complexzeros

#28 · Complex Analysis, Algebra & Topology › Singularities and Residues

“|f| bounded near an isolated singularity ⇒ pole” — false

Counterexample: f(z) = sin(z)/z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complexsingularities

#29 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“Every IVP has a unique solution” — false

Counterexample: y′ = y^{1/3}, y(0) = 0

Not Lipschitz at 0; y ≡ 0 and y = $(2 x /3)^{3/2}$ both solve it.

ODE

#30 · Probability & Statistics › Limit Theorems and Markov Chains

“Convergence in probability ⇒ almost sure convergence” — false

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probabilityconvergence

#31 · Probability & Statistics › Probability

“Uncorrelated ⇒ independent” — false

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probability

#32 · Analysis & Linear Algebra › The Real Line

“Cesàro means converge ⇒ the sequence converges” — false

Counterexample: aₙ = (−1)ⁿ

Partial averages → 0, sequence diverges.

sequences

#33 · Analysis & Linear Algebra › The Real Line

“aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L” — false

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sequences

#34 · Analysis & Linear Algebra › The Real Line

“limsup(aₙ + bₙ) = limsup aₙ + limsup bₙ” — false

Counterexample: aₙ = (−1)ⁿ, bₙ = (−1)ⁿ⁺¹

$a_{n} + b_{n} \equiv 0$ , so the left side is 0 while the right side is 1 + 1 = 2.

sequenceslimsup

#35 · Analysis & Linear Algebra › The Real Line

“Σaₙ converges ⇒ Σaₙ² converges” — false

Counterexample: aₙ = (−1)ⁿ/√n

Alternating series converges; squares give the harmonic series.

series

#36 · Analysis & Linear Algebra › The Real Line

“Ratio test inconclusive ⇒ root test inconclusive” — false

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series

#37 · Analysis & Linear Algebra › Continuity and Differentiation

“Bounded and continuous on ℝ ⇒ uniformly continuous” — false

Counterexample: f(x) = sin(x²)

$x_{n} = 2 π n, y_{n} = 2 π n + π /2$ satisfy | $x_{n} - y_{n}$ | → 0 while | $f (x_{n}) - f (y_{n})$ | = 1.

continuity

#38 · Analysis & Linear Algebra › Continuity and Differentiation

“Product of uniformly continuous functions is uniformly continuous” — false

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continuity

#39 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic and minimal polynomial ⇒ similar” — false

Counterexample: 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

Both have $χ = x^{4}$ and $m = x^{2}$ , but different numbers of blocks (ranks 2 vs 1).

linear algebracanonical forms

#40 · Analysis & Linear Algebra › Vector Spaces and Linear Maps

“AB and BA have the same minimal polynomial” — false

Counterexample: A = [[0,1],[0,0]], B = [[0,0],[0,1]]

AB = A has minimal polynomial $x^{2}$ , BA = 0 has x. The characteristic polynomials do agree.

linear algebra

#41 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Commuting matrices are simultaneously diagonalisable” — false

Counterexample: A = B = [[0,1],[0,0]]

They commute but neither is diagonalisable. Need each to be diagonalisable first.

linear algebra

#42 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Real matrix diagonalisable over ℂ ⇒ diagonalisable over ℝ” — false

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linear algebra

#43 · Analysis & Linear Algebra › Continuity and Differentiation

“If lim f/g exists (0/0 form) then lim f′/g′ exists” — false

Counterexample: f(x) = x² sin(1/x), g(x) = x as x → 0

f/g = x sin(1/x) → 0, but f′/g′ = 2x sin(1/x) − cos(1/x) has no limit. L'Hôpital goes one way only.

differentiation

#44 · Analysis & Linear Algebra › Continuity and Differentiation

“f′(x₀) = 0 and f′ changes sign nowhere near x₀ ⇒ f is constant near x₀” — false

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differentiation

#45 · Analysis & Linear Algebra › Integration

“|f| Riemann integrable ⇒ f Riemann integrable” — false

Counterexample: f = 1 on ℚ ∩ [0,1], −1 elsewhere

|f| ≡ 1 is integrable; f is discontinuous everywhere.

integration

#46 · Analysis & Linear Algebra › Integration

“A composition of Riemann integrable functions is Riemann integrable” — false

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integration

#47 · Analysis & Linear Algebra › Sequences and Series of Functions

“fₙ → f uniformly ⇒ fₙ′ → f′” — false

Counterexample: fₙ(x) = sin(nx)/n

Converges uniformly to 0, derivatives cos(nx) do not converge.

uniform convergence

#48 · Analysis & Linear Algebra › Sequences and Series of Functions

“fₙ → 0 pointwise on [0,1] ⇒ ∫₀¹ fₙ → 0” — false

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uniform convergenceintegration

#49 · Analysis & Linear Algebra › Sequences and Series of Functions

“Σaₙxⁿ → L as x → 1⁻ ⇒ Σaₙ = L” — false

Counterexample: Σ(−1)ⁿxⁿ = 1/(1 + x) → 1/2

$\sum (- 1)^{n}$ diverges. Abel's theorem has no converse without a Tauberian condition.

seriespower series

#50 · Analysis & Linear Algebra › Sequences and Series of Functions

“A power series converges uniformly on its open disc of convergence” — false

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power seriesuniform convergence

#51 · Analysis & Linear Algebra › Metric Spaces

“d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point” — false

Counterexample: T(x) = x + 1/x on [1, ∞)

Distances strictly decrease but no contraction constant k < 1 exists; no fixed point.

completenessfixed point

#52 · Analysis & Linear Algebra › Metric Spaces

“A countable dense subset of ℝ can be a G_δ” — false

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completenessbaire

#53 · Analysis & Linear Algebra › Metric Spaces

“Closure of a path-connected set is path-connected” — false

Counterexample: Graph of sin(1/x) on (0,1]

Its closure is the topologist's sine curve.

connectedness

#54 · Analysis & Linear Algebra › Metric Spaces

“Path components are closed” — false

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connectednesstopology

#55 · Complex Analysis, Algebra & Topology › Groups

“If d divides |G| then G has a subgroup of order d” — false

Counterexample: A₄ has order 12 but no subgroup of order 6

groups

#56 · Complex Analysis, Algebra & Topology › Groups

“o(ab) is finite whenever o(a) and o(b) are” — false

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groups

#57 · Complex Analysis, Algebra & Topology › Groups

“H ⊴ K and K ⊴ G imply H ⊴ G” — false

Counterexample: ⟨(12)(34)⟩ ⊴ V₄ ⊴ A₄

Conjugating (12)(34) by the 3-cycle (123) gives (13)(24) ∉ ⟨(12)(34)⟩.

groups

#58 · Complex Analysis, Algebra & Topology › Groups

“If every subgroup of G is normal then G is abelian” — false

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groups

#59 · Complex Analysis, Algebra & Topology › Groups

“Aₙ is simple for every n ≥ 3” — false

Counterexample: A₄ has the normal subgroup V₄ = {e, (12)(34), (13)(24), (14)(23)}

groupspermutations

#60 · Complex Analysis, Algebra & Topology › Groups

“A group with trivial centre cannot have order a prime power” — false

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groupsp-groups

#61 · Complex Analysis, Algebra & Topology › Groups

“G/Z(G) can be cyclic and non-trivial” — false

Counterexample: Impossible — if G/Z is cyclic then G is abelian, so G/Z is trivial

This 'no counterexample exists' fact is itself examined: |G/Z| is never prime.

groups

#62 · Complex Analysis, Algebra & Topology › Groups

“A group of order pq (p < q) is always cyclic” — false

Counterexample: S₃ of order 6 = 2·3

Non-abelian because 2 | 3 − 1. The cyclic conclusion needs p ∤ q − 1.

groupssylow

#63 · Complex Analysis, Algebra & Topology › Groups

“(ℤ/2^kℤ)* is cyclic for every k” — false

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groupsabelian

#64 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every prime ideal is maximal” — false

Counterexample: (X) in ℤ[X], or (0) in any integral domain that is not a field

$Z [X] / (X) ≅ Z$ is a domain but not a field. In a PID the implication does hold for non-zero primes.

rings

#65 · Complex Analysis, Algebra & Topology › Rings and Fields

“In every integral domain, irreducible implies prime” — false

Counterexample: 2 in ℤ[√−5]

2 is irreducible (no element of norm 2) but divides $(1 + \sqrt - 5) (1 - \sqrt - 5) = 6$ without dividing either factor.

rings

#66 · Complex Analysis, Algebra & Topology › Rings and Fields

“R a PID implies R[X] is a PID” — false

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rings

#67 · Complex Analysis, Algebra & Topology › Rings and Fields

“A polynomial irreducible over ℚ is irreducible modulo some prime” — false

Counterexample: x⁴ + 1

Irreducible over $Q$ , yet reducible modulo every prime.

polynomials

#68 · Complex Analysis, Algebra & Topology › Rings and Fields

“A degree-4 polynomial with no rational roots is irreducible over ℚ” — false

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polynomials

#69 · Complex Analysis, Algebra & Topology › Rings and Fields

“An algebraic extension is a finite extension” — false

Counterexample: The field of all algebraic numbers Q̄ over ℚ

Every element is algebraic, but the extension has infinite degree (it contains $Q (2^{1/ n})$ for every n).

fields

#70 · Complex Analysis, Algebra & Topology › Rings and Fields

“Every extension of degree n has a Galois group of order n” — false

Counterexample: ℚ(∛2)/ℚ has degree 3 but only the identity automorphism

The extension is not normal — the other cube roots are not real. Order = degree only for Galois extensions.

fieldsgalois

#71 · Complex Analysis, Algebra & Topology › Topology

“cl(A ∩ B) = cl(A) ∩ cl(B)” — false

Counterexample: A = ℚ, B = ℝ∖ℚ in ℝ

cl(A ∩ B) = cl(∅) = ∅ but cl A ∩ cl $B = R$ .

topology

#72 · Complex Analysis, Algebra & Topology › Topology

“A product of normal spaces is normal” — false

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topologyseparation

#73 · Complex Analysis, Algebra & Topology › Topology

“A continuous bijection is a homeomorphism” — false

Counterexample: id : (ℝ, discrete) → (ℝ, usual), or t ↦ (cos t, sin t) from [0, 2π) to S¹

Needs compact domain and Hausdorff codomain.

topology

#74 · Complex Analysis, Algebra & Topology › Topology

“Countably compact implies compact” — false

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topologycompactness

#75 · Complex Analysis, Algebra & Topology › Topology

“The closed unit ball of a normed space is compact” — false

Counterexample: The unit ball of ℓ²

Riesz: compactness of the ball holds exactly in finite dimensions.

topologycompactness

#76 · Complex Analysis, Algebra & Topology › Topology

“Connected components are open” — false

Counterexample: ℚ with the usual topology

Components are singletons, which are not open. They are open exactly when the space is locally connected.

topologyconnectedness

#77 · Complex Analysis, Algebra & Topology › Topology

“A totally disconnected space is discrete” — false

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topology

#78 · Complex Analysis, Algebra & Topology › Topology

“A compact T₁ space is Hausdorff” — false

Counterexample: An infinite set with the cofinite topology

Compact and $T_{1}$ but any two non-empty open sets intersect.

topology

#79 · Complex Analysis, Algebra & Topology › Analytic Functions

“The Cauchy–Riemann equations at a point imply complex differentiability there” — false

Counterexample: f(z) = (z̄)²/z for z ≠ 0, f(0) = 0

CR hold at 0, but the difference quotient along z = t(1+i) differs from the one along the real axis.

complexcauchy-riemann

#80 · Complex Analysis, Algebra & Topology › Analytic Functions

“A harmonic function on any domain has a harmonic conjugate” — false

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complexharmonic

#81 · Complex Analysis, Algebra & Topology › Analytic Functions

“Zeros of a non-constant holomorphic function cannot accumulate” — false

Counterexample: sin(1/z) on ℂ∖{0}

Zeros $1/ (nπ)$ accumulate at 0, which lies outside the domain. Inside the domain zeros are always isolated.

complexzeros

#82 · Complex Analysis, Algebra & Topology › Analytic Functions

“C^∞ implies analytic” — false

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complexanalyticity

#83 · Complex Analysis, Algebra & Topology › Cauchy Theory

“If f is holomorphic on the domain enclosed by γ except possibly at isolated points, and ∮_γ f = 0, then f has no singularity inside” — false

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complexcontour integration

#84 · Complex Analysis, Algebra & Topology › Cauchy Theory

“Cauchy's theorem applies to any domain without singularities of f” — false

Counterexample: f(z) = 1/z on the annulus ½ < |z| < 2

f is holomorphic there but $\oint_{∣ z ∣ = 1}$ dz $/ z = 2 π i \neq = 0$ — the annulus is not simply connected.

complex

#85 · Complex Analysis, Algebra & Topology › Cauchy Theory

“A bounded holomorphic function on an unbounded domain is constant” — false

Counterexample: f(z) = eᶻ on {Re z < 0}

|eᶻ| < 1 there, but f is not constant. Liouville needs the whole plane.

complexliouville

#86 · Complex Analysis, Algebra & Topology › Cauchy Theory

“An entire function omitting one value is constant” — false

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complexliouville

#87 · Complex Analysis, Algebra & Topology › Cauchy Theory

“|f| attains its minimum on the boundary for holomorphic f” — false

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complex

#88 · Complex Analysis, Algebra & Topology › Singularities and Residues

“If |f| is bounded near an isolated singularity, the singularity is a pole” — false

Counterexample: f(z) = sin z / z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complexsingularities

#89 · Complex Analysis, Algebra & Topology › Singularities and Residues

“A function has a unique Laurent expansion about a point” — false

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complexlaurent

#90 · Complex Analysis, Algebra & Topology › Singularities and Residues

“∮_γ f = 0 implies f is holomorphic inside γ” — false

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complexresidues

#91 · Complex Analysis, Algebra & Topology › Singularities and Residues

“A small indentation around a simple pole contributes 2πi·Res” — false

Counterexample: The indentation at 0 when computing ∫₀^∞ sin x/x dx

A half-circle contributes $iπ \cdot$ Res; using $2 π i$ gives twice the right answer.

complexresidues

#92 · Complex Analysis, Algebra & Topology › Zeros and Mappings

“f′(z) ≠ 0 everywhere implies f is injective” — false

Counterexample: f(z) = eᶻ on ℂ

f′ = eᶻ never vanishes, yet $f (z) = f (z + 2 π i)$ . Non-vanishing derivative gives only local injectivity.

complexinjectivity

#93 · Complex Analysis, Algebra & Topology › Zeros and Mappings

“ℂ and the unit disc are biholomorphic (both are simply connected)” — false

Counterexample: Liouville's theorem

A biholomorphism 𝔻 $\to C$ would invert to a bounded entire function. $C$ is the sole exception in the Riemann mapping theorem.

complexconformal

#94 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“A continuous right-hand side gives a unique solution” — false

Counterexample: y′ = y^{1/3}, y(0) = 0

Not Lipschitz at 0: y ≡ 0 and y = $(2 x /3)^{3/2}$ both solve it, as do infinitely many hybrids.

ODE

#95 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“A locally unique solution exists for all time” — false

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ODE

#96 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“W(f, g) ≡ 0 implies f and g are linearly dependent” — false

Counterexample: f(x) = x², g(x) = x|x| on ℝ

The Wronskian vanishes identically but no constant multiple relates them. The implication holds only for solutions of a common linear ODE.

ODEwronskian

#97 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“Every boundary value problem has a Green's function” — false

Counterexample: y″ + π²y = f on [0,1] with y(0) = y(1) = 0

$λ = π^{2}$ is an eigenvalue of the homogeneous problem, so the operator is not invertible and no Green's function exists.

ODEgreen

#98 · ODE, PDE & Applied Mathematics › Ordinary Differential Equations

“Linearisation determines stability at every equilibrium” — false

Counterexample: ẋ = −y − x³, ẏ = x − y³ at the origin

The linearisation is a centre (eigenvalues ±i), but $V = x^{2} + y^{2}$ gives V̇ $= - 2 (x^{4} + y^{4}) < 0$ : asymptotically stable. Non-hyperbolic equilibria escape Hartman–Grobman.

ODEstability

#99 · ODE, PDE & Applied Mathematics › Partial Differential Equations

“A first-order quasilinear Cauchy problem with smooth data has a global smooth solution” — false

Counterexample: u uₓ + u_y = 0 with u(x, 0) = −x

Characteristics $x = x_{0} (1 - y)$ all meet at y = 1: the solution blows up in gradient (a shock).

PDE

#100 · ODE, PDE & Applied Mathematics › Partial Differential Equations

“A PDE has a single type throughout its domain” — false

Counterexample: The Tricomi equation y uₓₓ + u_yy = 0

Elliptic in the upper half-plane, hyperbolic in the lower, parabolic on the axis.

PDE

#101 · ODE, PDE & Applied Mathematics › Partial Differential Equations

“All the classical equations satisfy a maximum principle” — false

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PDE

#102 · ODE, PDE & Applied Mathematics › Numerical Analysis

“Newton's method converges quadratically to any root” — false

Counterexample: f(x) = x² at the root 0

The root is double: $x_{n}$ ₊ $_{1} = x_{n} /2$ , only linear convergence.

numerical

#103 · ODE, PDE & Applied Mathematics › Numerical Analysis

“Newton's method always converges from any starting point” — false

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numerical

#104 · ODE, PDE & Applied Mathematics › Numerical Analysis

“Increasing the number of equally spaced interpolation nodes improves the approximation” — false

Counterexample: Runge's function 1/(1 + 25x²) on [−1, 1]

The interpolants diverge near the endpoints as $n \to \infty$ . Chebyshev nodes restore convergence.

numerical

#105 · ODE, PDE & Applied Mathematics › Numerical Analysis

“A convergent method is stable for any step size” — false

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numericalstability

#106 · ODE, PDE & Applied Mathematics › Calculus of Variations

“A weak minimum of a functional is a strong minimum” — false

Counterexample: J[y] = ∫₀^π (1 − (y′)²)y² dx at y ≡ 0

In the $C^{1} -$ ball the integrand is non-negative so J ≥ 0; in the $C^{0} -$ ball a steeply oscillating small y makes J negative.

calculus of variations

#107 · ODE, PDE & Applied Mathematics › Linear Integral Equations

“Every integral equation of the second kind has a unique solution” — false

Counterexample: y(x) − 3∫₀¹ tx y(t)dt = eˣ

$λ = 3$ is an eigenvalue of the kernel tx, and eˣ is not orthogonal to t — no solution exists.

integral equations

#108 · ODE, PDE & Applied Mathematics › Linear Integral Equations

“Volterra equations have eigenvalues just like Fredholm equations” — false

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integral equations

#109 · ODE, PDE & Applied Mathematics › Classical Mechanics

“The generalised momentum ∂L/∂q̇ always equals mass × velocity” — false

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mechanics

#110 · ODE, PDE & Applied Mathematics › Classical Mechanics

“The Hamiltonian always equals the total energy” — false

Counterexample: A bead on a wire rotating at forced angular velocity ω

The constraint is time-dependent, so H is conserved but differs from T + V.

mechanics

#111 · Probability & Statistics › Probability

“Pairwise independent events are mutually independent” — false

Counterexample: Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree

Each pair is independent, but P(A∩B∩C) = 1/4 ≠ 1/8 = P(A)P(B)P(C).

probability

#112 · Probability & Statistics › Probability

“Every random variable has a moment generating function” — false

Counterexample: The standard Cauchy distribution

E[ $e^{tX}] = \infty$ for every t ≠ 0; even E|X| is infinite. Its characteristic function $e^{- ∣ t ∣}$ exists.

probability

#113 · Probability & Statistics › Probability

“A mixture of two distributions is a linear combination of the variables” — false

Counterexample locked — unlock with Notes + PYQ

probability

#114 · Probability & Statistics › Probability

“The maximum of independent exponentials is exponential” — false

Counterexample: max(X₁, X₂) with Xᵢ ~ Exp(1)

The minimum is exponential (rate $λ_{1} + λ_{2})$ ; the maximum has CDF (1 − $e^{- x})^{2}$ , which is not exponential.

probabilitydistributions

#115 · Probability & Statistics › Probability

“If X and Y are each normal and uncorrelated then they are independent” — false

Counterexample: X ~ N(0,1), ε = ±1 with probability ½ independent of X, Y = εX

Y is N(0,1), Cov(X,Y) = 0, but |X| = |Y| always. The pair is not jointly normal.

probability

#116 · Probability & Statistics › Probability

“Uncorrelated implies independent” — false

Counterexample locked — unlock with Notes + PYQ

probability

#117 · Probability & Statistics › Limit Theorems and Markov Chains

“Convergence in probability implies almost sure convergence” — false

Counterexample: The typewriter sequence on [0,1]

$P (X_{n} \neq = 0) \to 0$ but every $ω$ is hit infinitely often, so there is no a.s. limit.

probabilityconvergence

#118 · Probability & Statistics › Limit Theorems and Markov Chains

“Xₙ → 0 almost surely implies E[Xₙ] → 0” — false

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probabilityconvergence

#119 · Probability & Statistics › Limit Theorems and Markov Chains

“An irreducible chain with a stationary distribution converges to it” — false

Counterexample: The two-state chain that swaps deterministically (period 2)

$π = ($ ½, ½) is stationary and unique, but $p_{nij}$ oscillates between 0 and 1. Aperiodicity is required.

probabilitymarkov

#120 · Probability & Statistics › Limit Theorems and Markov Chains

“A recurrent chain has a stationary distribution” — false

Counterexample locked — unlock with Notes + PYQ

probabilitymarkov

#121 · Probability & Statistics › Estimation

“A sufficient statistic is complete” — false

Counterexample: X₍ₙ₎ (or (X₍₁₎, X₍ₙ₎)) for Uniform(−θ, θ)

Sufficient but not complete: symmetry gives non-zero functions with zero expectation.

statistics

#122 · Probability & Statistics › Estimation

“The Cramér–Rao bound is attained by the UMVUE” — false

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statistics

#123 · Probability & Statistics › Estimation

“The MLE is unbiased” — false

Counterexample: σ̂² = (1/n)Σ(Xᵢ − X̄)² for N(μ, σ²), or X₍ₙ₎ for Uniform(0,θ)

Both underestimate systematically; $E [X_{(n)}] = n θ / (n + 1)$ .

statistics

#124 · Probability & Statistics › Estimation

“The MLE is unique” — false

Counterexample locked — unlock with Notes + PYQ

statistics

#125 · Probability & Statistics › Hypothesis Testing

“A UMP test exists for every testing problem” — false

Counterexample: H₀: μ = 0 vs H₁: μ ≠ 0 for N(μ, 1)

The MP test for $μ > 0$ rejects for large X̄, for $μ < 0$ for small X̄; no single test is best against both.

statisticstesting

#126 · Probability & Statistics › Hypothesis Testing

“Any interval of the form [X̄ − (S/√n)t, ∞) with a 90% quantile t is a 90% confidence interval” — false

Counterexample: t = t_{n−1,0.9} under the convention t_{m,α} = (1−α)-th quantile

That t is the 10th percentile, so the interval has coverage 10%, not 90%.

statistics

#127 · Probability & Statistics › Linear Models and Multivariate

“OLS is the BLUE in every linear model” — false

Counterexample: Yᵢ = βxᵢ + εᵢ with Var(εᵢ) = σ²xᵢ²

Gauss–Markov assumes constant variance; here weighted least squares (the mean of $Y_{i} / x_{i})$ has smaller variance.

statisticsregression

#128 · Probability & Statistics › Linear Models and Multivariate

“Every parameter in the one-way ANOVA model is estimable” — false

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statisticsanova

#129 · Probability & Statistics › Linear Models and Multivariate

“If every marginal is normal then the vector is multivariate normal” — false

Counterexample: X ~ N(0,1) and Y = εX with ε = ±1 independent

Both marginals are N(0,1) but X + Y is 0 half the time — not normal, so the pair is not jointly normal.

statisticsmultivariate

#130 · Probability & Statistics › Sampling and Design of Experiments

“Systematic sampling is always at least as efficient as SRS” — false

Counterexample: A population with a periodic pattern of period k, sampled every k-th unit

Every sampled unit falls at the same phase, so the sample can be maximally unrepresentative.

statisticssampling

#131 · Analysis & Linear Algebra › The Real Line

“Every ordered field is Archimedean” — false

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real analysis

#132 · Analysis & Linear Algebra › Integration

“If ∫₀^∞ f converges then f(x) → 0” — false

Counterexample: f with a spike of height n and width 2/n³ at each integer n

The total area is finite but f is unbounded, so it does not tend to 0.

integration

#133 · Analysis & Linear Algebra › Sequences and Series of Functions

“A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence” — false

Counterexample: fₙ(x) = xⁿ

Bounded by 1, but the pointwise limit is discontinuous so no subsequence converges uniformly. Equicontinuity is the missing hypothesis.

uniform convergencecompactness

#134 · Analysis & Linear Algebra › Functions of Several Variables

“If all partial derivatives exist at a point then f is continuous there” — false

Counterexample: f(x,y) = xy/(x²+y²), f(0,0) = 0

Both partials are 0 at the origin, but f = ½ along y = x, so f is not continuous.

multivariable

#135 · Analysis & Linear Algebra › Functions of Several Variables

“Mixed partial derivatives are always equal” — false

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multivariable

#136 · Analysis & Linear Algebra › Functions of Several Variables

“A C¹ map with everywhere non-zero Jacobian is injective” — false

Counterexample: f(x,y) = (eˣ cos y, eˣ sin y) on ℝ²

The Jacobian determinant is $e^{2 x}$ ≠ 0, but $f (x, y) = f (x, y + 2 π)$ . Invertibility is only local.

multivariable

#137 · Analysis & Linear Algebra › Metric Spaces

“An arbitrary intersection of open sets is open” — false

Counterexample: ∩_{n≥1}(−1/n, 1/n) = {0}

Only finite intersections preserve openness.

topologyreal analysis

#138 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“A set of measure zero is countable” — false

Counterexample: The Cantor set

Uncountable, yet measure zero.

measure

#139 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“A nowhere dense set has measure zero” — false

Counterexample locked — unlock with Notes + PYQ

measure

#140 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“Pointwise convergence implies convergence of the integrals” — false

Counterexample: fₙ = n·1_{(0,1/n)} on [0,1]

$f_{n} \to 0$ pointwise but $\int f_{n} = 1$ always. Domination or monotonicity is essential.

measureintegration

#141 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“Iterated integrals of a measurable function always agree” — false

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measure

#142 · Analysis & Linear Algebra › Lebesgue Measure and Integration

“L¹[0,1] ⊆ L²[0,1]” — false

Counterexample: f(x) = 1/√x

$\int f = 2 < \infty$ but $\int f^{2} = \int$ dx $/ x = \infty$ . On a finite measure space the inclusion runs the other way.

measurelp

#143 · Analysis & Linear Algebra › Vector Spaces and Linear Maps

“An injective linear operator on a vector space is surjective” — false

Counterexample: The right shift on ℓ²: (x₁, x₂, …) ↦ (0, x₁, x₂, …)

Injective but misses everything with a non-zero first coordinate. Rank–nullity needs finite dimension.

linear algebra

#144 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Every real matrix has a Jordan form over ℝ” — false

Counterexample: The rotation [[0,−1],[1,0]]

Its eigenvalues ±i are not real; over $R$ one uses the real Jordan form with a 2×2 rotation block.

linear algebra

#145 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Two matrices with the same characteristic and minimal polynomials are similar” — false

Counterexample locked — unlock with Notes + PYQ

linear algebra

#146 · Analysis & Linear Algebra › Inner Product Spaces and Forms

“A real matrix with all real eigenvalues is orthogonally diagonalisable” — false

Counterexample: [[1,1],[0,1]]

Eigenvalue 1 twice but only one eigenvector; orthogonal diagonalisability requires symmetry.

linear algebra

#147 · Analysis & Linear Algebra › Inner Product Spaces and Forms

“det A > 0 implies A is positive definite” — false

Counterexample: A = diag(−1, −1)

det = 1 > 0 but both eigenvalues are negative. All leading principal minors must be positive.

linear algebraquadratic forms

#148 · Analysis & Linear Algebra › Determinants and Matrix Tricks

“There exist matrices with AB − BA = I” — false

Counterexample: Impossible over ℝ or ℂ in finite dimensions

trace(AB − BA) = 0 but trace(I) = n ≠ 0. (It is possible for unbounded operators — the Heisenberg relation.)

linear algebra