# Category:Proofs by Contradiction

## Pages in category "Proofs by Contradiction"

The following 200 pages are in this category, out of 414 total.

(previous page) (next page)### A

- Abelian Group Factored by Prime
- Absolute Value of Components of Complex Number no greater than Root 2 of Modulus
- Additive Group and Multiplicative Group of Field are not Isomorphic
- Template:AimForCont
- Alexander's Compactness Theorem
- Alexandroff Extension of Rational Number Space is Connected
- Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2
- Algebraic Element of Field Extension is Root of Unique Monic Polynomial of Minimal Degree
- Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles
- Annihilating Polynomial of Minimal Degree is Irreducible
- Antiassociative Operation has no Idempotent Elements
- Antiperiodic Element is Multiple of Antiperiod
- Archimedean Principle/Variant
- Arens-Fort Space is not Countably Compact
- Axiom of Choice Implies Zorn's Lemma
- Axiom of Choice Implies Zorn's Lemma/Proof 2

### B

- Backwards Induction
- Beatty's Theorem
- Beatty's Theorem/Proof 1
- Beatty's Theorem/Proof 2
- Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')
- Between two Rational Numbers exists Irrational Number/Lemma 1
- Between two Rational Numbers exists Irrational Number/Lemma 2
- Bolzano-Weierstrass Theorem/General Form
- Bounded Above Subset of Real Numbers/Examples/Open Interval from 0 to 1
- Bounded Below Subset of Real Numbers/Examples/Open Interval from 0 to 1
- Brouwer's Fixed Point Theorem/One-Dimensional Version
- Brouwer's Fixed Point Theorem/One-Dimensional Version/Proof Using Connectedness
- Bézout's Lemma/Euclidean Domain

### C

- Cantor's Theorem
- Cantor's Theorem/Proof 1
- Cardinality of Mapping
- Center of Group of Order Prime Cubed
- Characterisation of Non-Archimedean Division Ring Norms/Corollary 2
- Characterization of Closure by Open Sets
- Characterizing Property of Infimum of Subset of Real Numbers
- Characterizing Property of Supremum of Subset of Real Numbers
- Circle is Bisected by Diameter
- Circle is Bisected by Diameter/Proof 1
- Clopen Set contains Components of All its Points
- Closed Extension Topology is not Hausdorff
- Closed Set/Complex Analysis/Examples/Closed Unit Circle
- Closed Topologist's Sine Curve is not Path-Connected
- Closed Unit Interval is not Countably Infinite Union of Disjoint Closed Sets
- Compact First-Countable Space is Sequentially Compact
- Compact Hausdorff Topology is Maximally Compact
- Compact Hausdorff Topology is Minimal Hausdorff
- Compact Sets in Countable Complement Space
- Compact Subspace of Linearly Ordered Space/Lemma
- Compact Subspace of Metric Space is Sequentially Compact in Itself
- Comparison Test for Convergence of Power Series
- Complement of Prime Ideal of Ring is Multiplicatively Closed
- Complement of Symmetric Relation
- Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars
- Complex Numbers cannot be Totally Ordered
- Complex Numbers cannot be Totally Ordered/Proof 1
- Complex Numbers cannot be Totally Ordered/Proof 2
- Complex Numbers cannot be Totally Ordered/Proof 3
- Component of Finite Union in Ultrafilter
- Condition for Open Extension Space to be Separable
- Condition for Repunits to be Coprime
- Condition for Uniqueness of Increasing Mappings between Tosets
- Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x
- Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x/Proof 1
- Congruence Modulo Integer/Examples/531 not equiv 1236 mod 7561
- Connected Subspace of Linearly Ordered Space
- Convergent Real Sequence has Unique Limit
- Convergent Real Sequence has Unique Limit/Proof 1
- Converse Hinge Theorem
- Coset/Examples/Subgroup of Infinite Cyclic Group
- Countable Complement Space is not Countably Compact
- Countable Complement Space is not First-Countable
- Countable Finite Complement Space is not Locally Path-Connected
- Countable Finite Complement Space is not Path-Connected
- Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point

### D

- Decomposition into Product of Power of 2 and Odd Integer is Unique
- Derivative of Exponential Function/Proof 5/Lemma
- Differential of Differentiable Functional is Unique/Lemma
- Dihedral Group D4 is not Internal Group Product
- Dihedral Group D4 is not Internal Group Product/Proof 1
- Discrete Space is Compact iff Finite
- Discrete Subgroup of Hausdorff Group is Closed
- Disjoint Permutations Commute
- Distance from Subset of Real Numbers to Supremum
- Distance from Subset of Real Numbers to Supremum/Proof 1
- Divergent Sequence may be Bounded
- Divergent Sequence may be Bounded/Proof 1
- Division Theorem for Polynomial Forms over Field
- Division Theorem for Polynomial Forms over Field/Proof 1
- Division Theorem/Positive Divisor/Positive Dividend/Uniqueness
- Division Theorem/Positive Divisor/Positive Dividend/Uniqueness/Proof 1
- Division Theorem/Positive Divisor/Uniqueness
- Division Theorem/Positive Divisor/Uniqueness/Proof 1
- Double Pointed Topology is not T0
- Double Pointed Topology is not T0/Proof 2

### E

- Element of Principal Ideal Domain is Finite Product of Irreducible Elements
- Equal Alternate Interior Angles implies Parallel Lines
- Equivalence of Definitions of Antisymmetric Relation
- Equivalence of Definitions of Ceiling Function
- Equivalence of Definitions of Compact Topological Space
- Equivalence of Definitions of Connected Topological Space
- Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space
- Equivalence of Definitions of Countably Compact Space
- Equivalence of Definitions of Floor Function
- Equivalence of Definitions of Infinite Cyclic Group
- Equivalence of Definitions of Infinite Order Element
- Equivalence of Definitions of Irreducible Space
- Equivalence of Definitions of Irreducible Space/3 iff 7
- Equivalence of Definitions of Order of Group Element
- Equivalence of Definitions of Prime Ideal of Commutative and Unitary Ring
- Equivalence of Definitions of Prime Number
- Equivalence of Definitions of Real Inverse Hyperbolic Cosine
- Equivalence of Definitions of Supremum of Real-Valued Function
- Equivalence of Definitions of T2 Space
- Equivalence of Definitions of Totally Separated Space
- Equivalence of Definitions of Ultraconnected Space
- Equivalence of Definitions of Ultraconnected Space/1 iff 3
- Equivalence of Definitions of Ultrafilter on Set
- Equivalence of Definitions of Ultrafilter on Set/Definition 1 iff Definition 3
- Equivalence of Definitions of Ultrafilter on Set/Equivalence of Definitions 1, 2 and 3
- Equivalence of Well-Ordering Principle and Induction
- Equivalence of Well-Ordering Principle and Induction/Proof/PCI implies WOP
- Equivalence of Well-Ordering Principle and Induction/Proof/WOP implies PFI
- Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class of Prime
- Error Correction Capability of Linear Code
- Euclid's Lemma for Prime Divisors/General Result
- Euclid's Lemma for Prime Divisors/General Result/Proof 3
- Euclid's Lemma for Unique Factorization Domain
- Euclid's Theorem/Corollary 2
- Euclid's Theorem/Corollary 2/Proof 1
- Euclid's Theorem/Corollary 2/Proof 2
- Euler's Number is Irrational
- Euler's Number is Transcendental
- Euler's Number is Transcendental/Proof 1
- Even Integer Plus 5 is Odd
- Even Integer Plus 5 is Odd/Proof by Contradiction
- Exclusive Or/Examples/One of Five Statements is True
- Existence of Minimal Uncountable Well-Ordered Set
- Existence of Minimal Uncountable Well-Ordered Set/Proof Without Using Choice
- Existence of Prime between Prime and Factorial
- Exponential of Rational Number is Irrational
- Exponential of Real Number is Strictly Positive
- Exponential of Real Number is Strictly Positive/Proof 5
- Exponential of Real Number is Strictly Positive/Proof 5/Lemma
- Expression for Integer as Product of Primes is Unique
- Expression for Integer as Product of Primes is Unique/Proof 1
- Expression for Integer as Product of Primes is Unique/Proof 2
- Expression for Integer as Product of Primes is Unique/Proof 3
- Expression of Vector as Linear Combination from Basis is Unique
- Extreme Value Theorem/Real Function

### F

- Feit-Thompson Conjecture/Stronger
- Fibonacci Number is not Product of Two Smaller Fibonacci Numbers
- Fibonacci Prime has Prime Index except for 3
- Field with 4 Elements has only Order 2 Elements
- Field with 4 Elements has only Order 2 Elements/Proof 1
- Field with 4 Elements has only Order 2 Elements/Proof 2
- Finite Integral Domain cannot be Ordered
- Finite Integral Domain is Galois Field
- Finite Integral Domain is Galois Field/Proof 4
- Fortissimo Space is not First-Countable
- Fully T4 Space is T4 Space

### G

### H

### I

- Ideal Contained in Finite Union of Prime Ideals
- Idempotent Non-Trivial Quasigroup is Not a Loop
- If Double Integral of a(x, y)h(x, y) vanishes for any C^2 h(x, y) then C^0 a(x, y) vanishes
- Image of Intersection under Injection
- Image of Intersection under Injection/Proof 2
- Infimum is not necessarily Smallest Element
- Infimum is not necessarily Smallest Element/Proof
- Infimum of Set of Reciprocals of Positive Integers
- Infinite Number of Primes of form 4n - 1
- Infinite Particular Point Space is not Countably Paracompact
- Infinite Particular Point Space is not Metacompact
- Infinite Subset of Finite Complement Space Intersects Open Sets
- Integer is Expressible as Product of Primes
- Integer is Expressible as Product of Primes/Proof 1
- Integer is Expressible as Product of Primes/Proof 2
- Integer to Power of p-1 over 2 Modulo p
- Integer to Rational Power is Irrational iff not Integer or Reciprocal
- Integral Ideal is Set of Integer Multiples
- Inverse Tangent of i
- Irrational Number/Examples/Cube Root of 2
- Irrational Number/Examples/Square Root of 3
- Irrationality of Logarithm
- Irreducible Polynomial/Examples/8 x^3 - 6 x - 1 in Rationals
- Isolated Point of Closure of Subset is Isolated Point of Subset
- Isolated Point of Closure of Subset is Isolated Point of Subset/Proof 1