Pages Needing Proofs
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Dedekind Cuts 0% (0/21) page hidden
- Natural Numbers Satisfy Arithmetic Laws theorem
- Natural Numbers are Well-Ordered theorem
- Integer Pair Relation is an Equivalence Relation proposition
- Integer Addition and Multiplication are Well-Defined proposition
- Order on Integers is Well-Defined proposition
- Integers Form an Ordered Integral Domain theorem
- Rational Number Pair Relation is an Equivalence Relation proposition
- Rational Addition and Multiplication are Well-Defined proposition
- Order on Rationals is Well-Defined proposition
- Rationals Form an Ordered Field theorem
- Rational Upper Rays are Dedekind Cuts lemma
- Complement of a Dedekind Cut lemma
- Dedekind Cuts are Linearly Ordered by Containment lemma
- Union of Dedekind Cuts is a Dedekind Cut lemma
- Dedekind Cut Operation Closure lemma
- Dedekind Cut Approximation lemma
- Order Comparison with Rational Cuts lemma
- Dedekind Reals Form an Ordered Field theorem
- Greatest Lower Bound Property for Dedekind Reals theorem
- Least Upper Bound Property for Dedekind Reals theorem
- Rationals Embed into Dedekind Reals theorem
Lebesgue Measure 0% (0/13) page hidden
- Outer Measure is Additive when One Set is Open theorem
- Outer Measure is Additive when One Set is Closed theorem
- Borel Sets Can be Approximated from Below by Closed Sets theorem
- Outer Measure is Additive when One Set is Borel theorem
- There is a Non-Borel Set of Finite Outer Measure theorem
- Outer Measure is a Measure on Borel Sets theorem
- Equivalent Characterizations of Lebesgue Measurable Sets theorem
- Outer Measure is a Measure on Lebesgue Measurable Sets theorem
- Base Three Description of the Cantor Set theorem
- Basic Properties of the Cantor Set theorem
- Cantor Function theorem
- Cantor Set is Uncountable corollary
- Continuous Image of a Lebesgue Measurable Set Can be Nonmeasurable theorem
Measurable Spaces And Functions 0% (0/12) page hidden
- Sigma Algebras are Closed under Standard Set Operations proposition
- Smallest Sigma Algebra Containing a Collection proposition
- Algebra of Inverse Images proposition
- Inverse Image of a Composition proposition
- Ray Test for Measurability proposition
- Continuous Functions are Borel Measurable theorem
- Increasing Functions are Borel Measurable theorem
- Composition of Measurable Functions proposition
- Algebraic Operations with Measurable Functions proposition
- Pointwise Limit of Measurable Functions is Measurable theorem
- Extended Ray Test for Measurability proposition
- Infimum and Supremum of Measurable Functions proposition
Integration 0% (0/11) page hidden
- Integral of a Characteristic Function proposition
- Integral of a Simple Function proposition
- Integration of Nonnegative Functions is Order Preserving proposition
- Monotone Convergence Theorem theorem
- Additivity of Integration for Nonnegative Functions theorem
- Additivity of Integration theorem
- Bounded Convergence Theorem theorem
- Dominated Convergence Theorem theorem
- Riemann Integrable iff Continuous Almost Everywhere theorem
- Approximation in \( L ^ 1 \) by Simple Functions theorem
- Approximation in \( L ^ 1 \) by Continuous Functions theorem
Outer Measure 0% (0/9) page hidden
- Countable Sets Have Outer Measure Zero proposition
- Outer Measure Preserves Order proposition
- Outer Measure is Translation Invariant proposition
- Countable Subadditivity of Outer Measure proposition
- Heine-Borel Theorem theorem
- Outer Measure of a Closed Interval theorem
- Nontrivial Intervals are Uncountable corollary
- Outer Measure is Not Additive theorem
- No Countably Additive Translation Invariant Extension of Length to All Subsets theorem
Hilbert Space Theory 0% (0/7) page hidden
- Cauchy-Schwarz Inequality theorem
- Parallelogram Identity theorem
- Closest Point Projection onto Closed Convex Set theorem
- Orthogonal Decomposition theorem
- Bessel's Inequality theorem
- Parseval's Identity theorem
- Riesz Representation Theorem for Hilbert Spaces theorem
Metric And Banach Spaces 0% (0/7) page hidden
- Banach Fixed Point Theorem theorem
- Linear Map Continuous iff Bounded theorem
- Bounded Linear Maps Form a Banach Space theorem
- Baire Category Theorem theorem
- Open Mapping Theorem theorem
- Closed Graph Theorem theorem
- Uniform Boundedness Principle theorem
Convergence Of Measurable Functions 0% (0/7) page hidden
- Uniform Limit of Continuous Functions is Continuous theorem
- Egorov's Theorem theorem
- Approximation by Simple Functions theorem
- Luzin's Theorem theorem
- Continuous Extensions from Closed Subsets of the Real Line theorem
- Luzin's Theorem, Extension Version theorem
- Every Lebesgue Measurable Function is Almost Borel Measurable theorem
Monte Carlo 0% (0/7) page hidden
- Joint Density Factors as Marginal Times Conditional Density proposition
- Monte Carlo Estimator is Unbiased proposition
- Variance of an Average of Independent Identically Distributed Random Variables proposition
- Inversion Method Samples from the Target Distribution theorem
- Change of Variables for Density Functions theorem
- Stratified Sampling Does Not Increase Variance proposition
- Russian Roulette Estimator is Unbiased proposition
Riemann Integration 0% (0/6) page hidden
- Riemann Sum Inequalities under Refinement proposition
- Any Lower Riemann Sum is Below Any Upper Riemann Sum proposition
- Lower Riemann Integral is Below Upper Riemann Integral proposition
- Continuous Functions on Closed Bounded Intervals are Riemann Integrable theorem
- Bounds on a Riemann Integral proposition
- Interchanging Riemann Integral and Pointwise Limit theorem
Lp Spaces 0% (0/6) page hidden
- Holder's Inequality theorem
- Minkowski's Inequality theorem
- \( L ^ p \) is Complete theorem
- Density of Simple Functions in \( L ^ p \) theorem
- Density of Continuous Compactly Supported Functions in \( L ^ p \) theorem
- Riesz-Fischer Theorem theorem
Measures 0% (0/5) page hidden
- Measure Preserves Order and Set Difference proposition
- Countable Subadditivity of Measures proposition
- Measure of an Increasing Union theorem
- Measure of a Decreasing Intersection theorem
- Measure of a Union proposition
Product Measures 0% (0/5) page hidden
- Sections of Measurable Sets are Measurable proposition
- Tonelli's Theorem theorem
- Fubini's Theorem theorem
- Lebesgue Measure on Products theorem
- Change of Variables for Lebesgue Integrals theorem
Eulers Totient Function 0% (0/5) page hidden
- Bijection between Cops and G's proposition
- The GCD Sets form a Partition proposition
- Euler's theorem
- Sum over Coprimes Totient Identity proposition
- Sum over Divisors of the Totient function is the Identity lemma
Fourier Analysis 0% (0/4) page hidden
- Riemann-Lebesgue Lemma theorem
- Fejer's Theorem theorem
- Fourier Series Converges in \( L ^ 2 \) theorem
- Plancherel Identity for Fourier Series theorem
Operators On Hilbert Spaces 0% (0/4) page hidden
Differentiation 0% (0/4) page hidden
Real And Complex Measures 0% (0/4) page hidden
- Hahn Decomposition Theorem theorem
- Jordan Decomposition Theorem theorem
- Radon-Nikodym Theorem theorem
- Lebesgue Decomposition Theorem theorem
Probability Measures 0% (0/4) page hidden
Linear Transformations 0% (0/3) page hidden
- The Dot Product in Rn Forms an Inner Product proposition
- Schwarz Inequality proposition
- The Norm Satisfies the Triangle Inequality proposition
Dihedral Group 0% (0/2) page hidden
- Elements of a Dihedral Group proposition
- Caley Table for D _ 4 proposition
Span And Linear Independence 0% (0/2) page hidden
Compactness And Extreme Values 0% (0/2) page hidden
- The Continuous Image of a Compact set is Compact theorem
- Extreme Value theorem
Power Series 0% (0/2) page hidden
- Term by Term Operations on Series proposition
- Hadamard theorem
Uniform Convergence And Integration 0% (0/2) page hidden
- Integral Convergence theorem
- Derivative Convergence proposition
Merge Sort 0% (0/2) page hidden
Missing Number 0% (0/2) page hidden
- A Sorted Tuple Missing an Element Has a Change Point proposition
- Equal Number of Ones and Zeros proposition
Properties Of Continuous Functions 0% (0/1) page hidden
Combinatorics 0% (0/1) page hidden
- painting exercise
Review 0% (0/1) page hidden
- Nested Triangles with Recursion exercise
Master Theorem 0% (0/1) page hidden
- Master theorem
Electricity 0% (0/1) page hidden
- Metals are Conductors proposition
Riemann Zeta Function 0% (0/1) page hidden
- Zeta Function1 theorem
Introduction 0% (0/1) page hidden
- Union proposition
Tuples 13% (1/8) page hidden
- Length of a Zero Indexed Tuple proposition
- Converting from Regular to Zero Indexing proposition
- Reverse Cancels proposition
- sorted_asc Holds iff It is Sorted in Ascending Order corollary
- Ascending Tuple exercise
- Descending Tuple exercise
- The reverse of Ascending is Descending proposition
Floor And Ceiling 15% (2/13) page hidden
- Fractional Part of Floor Bound corollary
- Floor of a Non Integer is Smaller proposition
- Ceiling of a Non Integer is Greater proposition
- The floor of a Sum of a Real and an Integer proposition
- A Number is a Perfect Square if its Square Root is an Integer lemma
- Counting Squares with Floor corollary
- When Flooring Tells us Two Numbers Divide Eachother proposition
- Counting Multiples with Floor proposition
- Largest Prime Power Dividing the Factorial proposition
- Prime Factorization of the Factorial corollary
- Double Sum of Divisors Equation proposition
Compactness And Subsets Of C(K) 20% (1/5) page hidden
Arithmetic Functions 21% (3/14) page hidden
- Equation for the Divisor Counting Function proposition
- Equation for the Divisor Sum Function proposition
- The Base Function is Totally Multiplicative corollary
- When the Constant Function is Multiplicative proposition
- The Product of Two Multiplicative Functions is Multiplicative theorem
- The Quotient of Two Multiplicative Functions is Multiplicative theorem
- Two Multiplicative Functions are Equal if they Agree on Prime Powers theorem
- The Sum of a Multiplicative Function over Divisors of a Number is Multiplicative theorem
- The Dirichlet Convolution of two Multiplicative Functions is Multiplicative theorem
- The Dirichlet Identity is Completely Multiplicative corollary
- The Dirichlet Identity is an Identity corollary
Limits And Continuity 25% (1/4) page hidden
Expected Value 25% (1/4) page hidden
- Expectation is Linear proposition
- Variance as Squares of Expection corollary
- Variance is Not Linear proposition
Prime Ideals And Maximal Ideals 27% (3/11) page hidden
- Polynomial of Degree 2 or 3 is Irreducible Iff It has no Roots proposition
- Irreducible Either has GCD 1 or Divides any other Non-Constant Polynomial proposition
- Irreducibility Criterion with Prime Ideals theorem
- A Proper Ideal is Prime iff the Quotient Ring is a Domain proposition
- Maximal iff Quotient Ring is a Field proposition
- Maximal Implies Prime corollary
- In a PID every Non-Zero Prime Ideal is Maximal proposition
- Splits iff Roots corollary
Matrices 29% (2/7) page hidden
- The Packing and Unpacking Functions are Inverses proposition
- One by one Matrices are Isomorphic to R proposition
- Row and Column Matrices are Isomorphic to \( \mathbb{ R } ^ n \) proposition
- Nested Matrix Multiplication Unpacking lemma
- Matrix Multiplication with Rows and Columns proposition
Rings 33% (8/24) page hidden
- Two by Two Matrices with 0 In Bottom Left Form a Crone exercise
- The Integers form a Ring exercise
- Every Field is a Ring corollary
- The Integers Modulo N are a Ring exercise
- Polynomials Form a Ring with Ring Coefficients proposition
- Square Matrices form a Non-Commutative Ring exercise
- Zero Multiplication in a Ring proposition
- Additive Inverse equals Multiplication by One's Additive Inverse proposition
- Additive Inverse times Additive Inverse of One Yields Original proposition
- The set of One Element Forms a Ring proposition
- Polynomials Formed from a Domain Form a Domain corollary
- Subring Criterion proposition
- The complex numbers are a field extension of the reals exercise
- Field Extension Yields a Vector Space proposition
- The Kernel of a Ring Homomorphism is a Proper Ideal proposition
- A Ring Homomorphism is an Injection iff It's Kernel is Zero proposition
Rubiks Cube 33% (1/3) page hidden
Runtime 33% (1/3) page hidden
Graphs 33% (1/3) page hidden
- Sum of Degrees proposition
- A Disconnected Graph is the Union of Many Disjoint Connected Subgraphs proposition
Continuous Distributions 33% (1/3) page hidden
Polynomial Rings Over Fields 40% (2/5) page hidden
- GCD as a Linear Combination proposition
- Every Polynomial Ideal is Principal proposition
- Show that \( \mathbb{ Z } \) is a Principal Ideal Domain exercise
Square Free Integers 40% (2/5) page hidden
Legendre Symbol 50% (6/12)
- Eulers Criterion lemma
- Quadratic Residue iff Legendre Symbol is One corollary
- Obtaining a Solution to a Quadratic Congrugence mod p Squared from a Solution mod p proposition
- A Number is a Solution of a Quadratic Equation mod a Power of 2 iff a Power of 2 Minus the Number is as well lemma
- Only Numbers Congruent to 1 mod 8 Have Quadratic Residues mod a Power of 2 proposition
- A Quadratic Congruence mod a Power of 2 has 4 Solutions proposition
Probability Models 50% (4/8)
- Intersection as Conditional Probability corollary
- Symmetric Conditional Equation corollary
- Probability of Set Difference proposition
- Law of Total Probability theorem
Fields 50% (3/6)
- The Complex Numbers Form a Field proposition
- A Field is a Crone corollary
- A Crone with Multiplicative Inverses is a Field corollary
Convergence And Completeness 50% (3/6)
Symmetric Difference 50% (3/6)
- Set Equality through the Characteristic corollary
- Symmetric Difference Characterization proposition
- Characteristic Version of Symmetric Difference proposition
Recursion 50% (2/4)
- foldr Expression proposition
- foldl Expression proposition
Structures And Languages 50% (2/4)
- For All Makes a Variable not Free corollary
- Free Variable For All Easy exercise
Homomorphisms 50% (1/2)
- Homomorphisms carry Zero to Zero proposition
Subgroups 50% (1/2)
Uniform Continuity 50% (1/2)
- Every Lipschitz function is Uniformly Continuous proposition
Uniform Convergence And Continuity 50% (1/2)
Interval Scheduling 50% (1/2)
Syntax And Grammar 50% (1/2)
- Syntactically Valid Arithmetic exercise
Hilbert Spaces 50% (1/2)
- Uniform Boundedness Theorem theorem
Mobius Inversion 50% (1/2)
- The Dirichlet Inverse for 1 is the Inverse proposition
Limits Of Sets 50% (1/2)
Ideals 53% (8/15)
- Trivial Ideal corollary
- Every Crone is an Ideal corollary
- The Kernel of a Crone Homomorphism is a Proper Ideal proposition
- A Crone Homomorphism is an Injection iff \( \operatorname{ ker } \left( \phi \right) = \left\{ 0 _ R \right\} \) proposition
- An Ideal is a Normal Subgroup proposition
- Can't get to All Polynomials From a Generated Ideal exercise
- Quotient Remainder For Polynomials proposition
Breadth First Search 56% (5/9)
- Breadth First Search is Correct theorem
- BFS Searches Less Deep Vertices First proposition
- BFS Expands by Depth Layer corollary
- BFS Expands All Enqueued Vertices proposition
Closed And Open Subsets 59% (16/27)
- Finite Unions and Arbitrary Intersections are Closed proposition
- A Convergent Sequence With Finite Image Must Converge to a Point in Its Image lemma
- The Closure is Closed corollary
- Closed iff Equals Closure proposition
- The Closure of a Set is the Smallest Closed Set Containing It proposition
- The Open Ball is Open corollary
- Open iff Its Complement is Closed theorem
- The Arbitrary Union and Finite Intersection of Open sets is Open proposition
- Compact Implies Closed and Bounded lemma
- A Closed Subset of a Compact Set is Compact lemma
- Heine Borel theorem
Logic 60% (6/10)
- Exclusive Or Evaluates to True iff an odd number of arguments are True corollary
- Contrapositive proposition
- Forced Consequence proposition
- For Every, There Exists is a Function proposition
Uniform Cost Search 60% (3/5)
A Star 63% (5/8)
Primitive Roots 64% (7/11)
Discrete Distributions 64% (9/14)
- Expected Value of the Binomial Distribution corollary
- Probability Mass Function of the Binomial Distribution proposition
- \( T _ k = n \) Equivalence proposition
- Probability Mass Function of the Negative Binomial Distribution proposition
- Fundamental Properties of Covariance proposition
Quotient Rings 66% (21/32)
- The Natural Map is a Surjective Group Homomorphism corollary
- Quotient Ring Mod a Polynomial proposition
- Quotient Ring of Polynomial is Same as Its Remainder proposition
- Crone Homomorphism induces an Isomorphism to its Image theorem
- Polynomials Mod a Polynomial is a Field iff it's irreducible proposition
- Quintic Formula Examples exercise
- Number of Irreducible Polynomials exercise
- Splits Iff has all Roots corollary
- Irreducible and Monic with a Root has a Unique Smallest Degree lemma
- Irreducible Polynomial of Degree \( d \) Creates a Field Extension of Degree \( d \) corollary
- Constructing a Reduced Polynomial proposition
Vectors 67% (6/9)
- Dot Product Geometric proposition
- Norm Squared is the Dot Product corollary
- Cross Product Vector Component Formula proposition
Intermediate Value Theorem 67% (4/6)
- Intermediate Value theorem
- Intermediate Value for Paths corollary
Mean Value Theorem 67% (4/6)
- Sign of the Derivative Tells us about Monotonicity corollary
- Rolles and Zeros corollary
Normal Subgroups 67% (2/3)
- Normal Subgroup Test proposition
Weierstrass Approximation 67% (2/3)
- Weierstrass Approximation theorem
Counting Inversions 67% (2/3)
- \( \texttt{counting_inversions} \) is Correct proposition
Ford Fulkerson 67% (2/3)
- Max-Flow Min-Cut theorem
Interval Partitioning 67% (2/3)
- \( \texttt{counting_inversions} \) is Correct proposition
Summations 67% (2/3)
- Sum of Consecutive Integers theorem
Generalized Gcd And Lcm 67% (2/3)
- GCD of GCD's Is the Same as Their Union proposition
Random Variables 67% (2/3)
- Independence as a Product corollary
Quotient And Remainder 68% (13/19)
- Quotient Remainder using Quotient and Remainder Function corollary
- Specific Value of the Divisor Function corollary
- Specific Value of the Remainder Function corollary
- When the Remainder Function does Nothing proposition
- Repeated Application of the Remainder Function does Nothing proposition
- Adding Multiples of the Divisor Never changes the Remainder proposition
Algebra Of The Complex Plane 70% (7/10)
- extracting the real part of a complex number proposition
- modulus is greater than it's components proposition
- imaginary part distributes proposition
Complex Numbers 70% (7/10)
- extracting the real part of a complex number proposition
- modulus is greater than it's components proposition
- imaginary part distributes proposition
The Fundamental Theorem Of Calculus 71% (5/7)
- Fundamental Theorem of Calculus I theorem
- Fundamental Theorem of Calculus II theorem
Riemann Integration 73% (8/11)
- Upper Sum Decreases over Refinements proposition
- Lower Sum Decreases over Refinements proposition
- Riemann Integrable Characterizations theorem
Gcd And Lcm 73% (22/30)
- GCD's Characterization corollary
- Factoring GCD proposition
- Non Trivial Divisors Don't Change with the Sign Changes corollary
- Non Trivial Divisors is a Subset of 2 up to The Number corollary
- Common Divisors as the Intersection of the Non Trivial Divisors corollary
- Common Divisor Set is Bounded Above corollary
- No Common Non Trivial Divisors Implies Relatively Prime proposition
- Least Common Multiple Characterization corollary
Summations And Series 75% (15/20)
- Non-Negative Series Converges iff Partial Sums are Bounded Above proposition
- Monotone Increasing Sequence with a Subsequence which is Bounded Above Implies Entire Sequence is Bounded Above lemma
- Ratio Test proposition
- Leibniz Alternating Series Test proposition
- Every Rearrangment of an Absolutely Convergent Series converges to the same Limit theorem
Basics 75% (6/8)
- Pigeon Hole Principle proposition
- Formula For the \( n \)-th Catalan Number proposition
Symmetry Group 75% (3/4)
- The Symmetry Group is a Group proposition
Dirichlet Theorem 75% (3/4)
- Euler product of L-function theorem
Probability Measures 75% (3/4)
Binary Operations And Groups 76% (16/21)
- The Integers are a Group proposition
- The Integers with Addition are Abelian proposition
- Cartesian Product of Abelian Groups is Abelian proposition
- Finite Order Implies Powers Have Same Remainder proposition
- gcd generates the same theorem
Primes 79% (41/52)
- Fundamental Theorem of Algebra theorem
- Products as the Addition of Power Sequences corollary
- Max of the Prime Power Sequence is 0 iff They share no Prime Factors lemma
- Relatively Prime In terms of Prime Factorization proposition
- Subsequence of the sum of Two Disjoint Power Sequences Is Still a Disjoint Sum lemma
- Euclid lemma
- If a Prime Divides a Product it Divides at Least One of the Factors corollary
- A Divisor of a Product of Primes is a Product of Primes with Smaller Powers proposition
- GCD and LCM as Prime Powers proposition
- Division and Prime Power Equivalence proposition
- Korselt's Criterion corollary
Induction 80% (16/20)
- Sum of the Squares of Consecutive Integers proposition
- Multiplying by a non Negative Constant Retains Inequality lemma
- The Square of a Real Number is Non-Negative proposition
- Finite Induction proposition
Homomorphisms 80% (8/10)
Number Bases 80% (4/5)
The Galois Group 82% (14/17)
Quadratic Congruences 83% (5/6)
- Number of Solutions to a Quadratic Congruence proposition
Boundedness 86% (12/14)
Domains And Fields 86% (6/7)
- Field is a Ring with Units proposition
Limits 90% (26/29)
- Eventually Increasing Index Mapping Maintains Limit proposition
- limsup Exists iff the Sequence is Bounded Above proposition
- Inequality From Sequence to Limit proposition
Hausdorff Spaces 91% (10/11)
Absolute Value 91% (21/23)
- Maximum is Always Bigger than It's arguments proposition
- Triangle Inequality theorem
Congruence 92% (33/36)
- Congruence Commutes proposition
- Sum of Two Congruent Numbers are Congruent proposition
- Product of Two Congruent Numbers is Congruent proposition
Functions 92% (23/25)
- Invertible iff Bijective proposition
- When a Function Defined in Terms of a Function is a Function proposition