Pages Needing Proofs

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

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

Linearly Independent iff Unique Representation proposition
Linearly Depedent iff Zero has a Non-Trivial Representation corollary

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

\( \texttt{merge} \) is Correct lemma
\( \texttt{merge_sort} \) is Correct theorem

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

Sequential and Open Set Equivalences for Continuity theorem

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

Riemann Zeta Function 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

Compact Subsets of Continuous Functions with Compact Domain is Equicontinuous lemma
An Equicontinuous Family of Functions whose Domain is Compact is also Uniformly Equicontinuous proposition
A Bounded subset of Rm is Totally Bounded proposition
Arzela-Ascoli theorem

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

The Characteristic Function is Discontinuous on the Boundary exercise
Every Lipschitz Function is Continuous proposition
Base and Exponent in the Range 0 to 1 Inequality lemma

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

If a Sequence of Moves Yields the Identity then applying those Moves to the Physical Cube Means it is Solved proposition
corollary

Runtime 33% (1/3) page hidden

Squared Dominates Linear exercise
Lower Degree Polynomial plus Constant is Eventually Dominated by Higher Degree exercise

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

Probability of an Interval in the Uniform Distribution corollary
Probability of a Point is Zero for an Absolutely Continuous Random Variable proposition

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

Square Free Division Characterization corollary
Divisor of Square Free Product is Divisor of Other proposition
Power of a Square free Number Divides the Power of a Number Implies the Square Free Number Divides the Other proposition

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)

Convergence iff Limit gets arbitrarily Close to a Point corollary
A Sequence in Rn Converges to a Point iff Each Component Converges proposition
\( \mathbb{ R } ^ n \) is Complete theorem

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)

The Subgroup generated by S is the Smallest Subgroup containing S corollary

Uniform Continuity 50% (1/2)

Every Lipschitz function is Uniformly Continuous proposition

Uniform Convergence And Continuity 50% (1/2)

Uniform Convergence Equivalence In Terms of Lim Sup theorem

Interval Scheduling 50% (1/2)

Earliest Finish Time Adding Criterion lemma

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)

Half Closed Subset of Open for Smaller Value lemma

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)

Monontonicity implies Admissibility theorem
Later Expansions Imply Larger f-values Under Monotonic Heuristic proposition
First Expansion is Minimum Cost Uner Monotonic Heuristic proposition

Primitive Roots 64% (7/11)

Order of a Primitive Root mod the next Power proposition
When a Primitive Remains a Primitive Root mod the next Power corollary
Every Power of an odd Prime has a Primitive Root proposition
Classification of the Existence of Primitive Roots theorem

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)

Probability of a Random Variable being an Element of a Set using Inverse Image corollary

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
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)

Homomorphism Between a Group and it's Cartesian Product exercise

Number Bases 80% (4/5)

Base Plus Minus One To a Power is Congruent to Plus Minus One lemma

The Galois Group 82% (14/17)

Degree 1 iff Same as Fields proposition
Irreducible Polynomial Restricts Size of Galois Group proposition
Automorphism Takes Roots of a Factor to the Root of the Same Factor corollary

Quadratic Congruences 83% (5/6)

Number of Solutions to a Quadratic Congruence proposition

Boundedness 86% (12/14)

Supremum of Suprema Is the Same as Their Union proposition
A Number is Greater than the Other iff the Difference is Positive proposition

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)

Every Finite Set is Closed in a Hausdorff Space corollary

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

Everything 95% (19/20)

Bounded Linear Functional Integral lemma

Division 95% (21/22)

If a Number Divides another Then Given a Common Factor Their Quotients Divide Each Other proposition

Banach Spaces 96% (22/23)

The Operator Norm of a Bounded Operator Is The Smallest Number Satisfying an Inequality proposition