# Category:Proofs by Contradiction

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## Pages in category "Proofs by Contradiction"

The following 200 pages are in this category, out of 791 total.

(previous page) (next page)### 1

### A

- Abelian Group Factored by Prime
- Absolute Value is Many-to-One
- Absolute Value of Components of Complex Number no greater than Root 2 of Modulus
- Absolutely Convergent Product Does not Diverge to Zero
- Absolutely Convergent Product Does not Diverge to Zero/Proof 1
- Absolutely Convergent Product Does not Diverge to Zero/Proof 2
- Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space
- 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
- Alternative Definition of Ordinal in Well-Founded Theory
- Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles
- Annihilating Polynomial of Minimal Degree is Irreducible
- Anomalous Cancellation on 2-Digit Numbers
- Antiassociative Operation has no Idempotent Elements
- Antilexicographic Order/Examples/Unit Square with Open Side
- Antiperiodic Element is Multiple of Antiperiod
- Antireflexive and Transitive Relation is Asymmetric
- Archimedean Principle/Variant
- Area of Circle
- Area of Circle/Proof 3
- Arens-Fort Space is not Countably Compact
- Arens-Fort Space is not First-Countable
- Asymmetric Relation is Antireflexive
- Axiom of Choice Implies Zorn's Lemma
- Axiom of Choice Implies Zorn's Lemma/Proof 2
- Axiom of Pairing from Powers and Replacement

### B

- Backwards Induction
- Basic Universe has Infinite Number of Elements
- Basic Universe is not Set
- 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
- Boolean Prime Ideal Theorem/Extension Lemma
- 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
- Bounded Below Subset of Real Numbers/Examples/Real Numbers
- 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 (Strong Version)
- Cantor's Theorem (Strong Version)/Proof 2
- Cantor's Theorem/Proof 1
- Cardinal Equal to Collection of All Dominated Ordinals
- Cardinalities form Inequality implies Difference is Nonempty
- Cardinality of Mapping
- Cauchy's Inequality
- Cauchy's Inequality/Proof 1
- Center of Group of Order Prime Cubed
- Characterisation of Jacobson Radical
- Characterisation of Non-Archimedean Division Ring Norms/Corollary 2
- Characteristic of Field by Annihilator/Characteristic Zero
- Characteristic of Field by Annihilator/Prime Characteristic
- Characterization of Closure by Open Sets
- Characterizing Property of Infimum of Subset of Real Numbers
- Characterizing Property of Supremum of Subset of Real Numbers
- Chords do not Bisect Each Other
- Circle is Bisected by Diameter
- Circle is Bisected by Diameter/Proof 1
- Class has Subclass which is not Element
- Clopen Set contains Components of All its Points
- Closed Balls Centered on P-adic Number is Countable/Lemma
- Closed Bounded Subset of Real Numbers is Compact
- Closed Bounded Subset of Real Numbers is Compact/Proof 1
- Closed Extension Topology is not Hausdorff
- Closed Real Interval is Compact
- Closed Real Interval is Compact/Topological Space
- Closed Set of Uncountable Finite Complement Topology is not G-Delta
- 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
- Closure of Pointwise Operation on Algebraic Structure
- Compact First-Countable Space is Sequentially Compact
- Compact Hausdorff Topology is Maximally Compact
- Compact Hausdorff Topology is Minimal Hausdorff
- Compact Set of Rational Numbers is Nowhere Dense
- Compact Sets in Countable Complement Space
- Compact Subspace of Linearly Ordered Space
- Compact Subspace of Linearly Ordered Space/Lemma
- Compact Subspace of Linearly Ordered Space/Reverse Implication
- Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 2
- 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
- Complete and Totally Bounded Metric Space is Sequentially Compact
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 2
- Complex Numbers cannot be Extended to Algebra in Three Dimensions with Real Scalars
- Complex Numbers cannot be Ordered Compatibly with Ring Structure
- Complex Numbers cannot be Ordered Compatibly with Ring Structure/Proof 1
- Complex Numbers cannot be Ordered Compatibly with Ring Structure/Proof 2
- Complex Numbers cannot be Ordered Compatibly with Ring Structure/Proof 3
- Complex Numbers under Multiplication do not form Group
- Complex Sine Function is Unbounded
- Component of Finite Union in Ultrafilter
- Composition of 3 Mappings where Pairs of Mappings are Bijections
- Condition for Composite Relation with Inverse to be Identity
- Condition for Denesting of Square Root/Lemma
- Condition for Differentiable Functional to have Extremum
- Condition for Open Extension Space to be Separable
- Condition for Planes to be Parallel
- Condition for Relation to be Transitive and Antitransitive
- Condition for Repunits to be Coprime
- Condition for Straight Lines in Plane to be Parallel/General Equation
- 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
- Continuous Function on Closed Real Interval is Uniformly Continuous
- Continuous Function on Closed Real Interval is Uniformly Continuous/Proof 2
- Continuous Function on Compact Space is Bounded
- Continuous implies Increasing in Scott Topological Lattices
- Convergence of Odd and Even Subsequences to Same Limit
- Convergent Real Sequence has Unique Limit
- Convergent Real Sequence has Unique Limit/Proof 1
- Convergent Sequence in Hausdorff Space has Unique Limit
- Convergent Sequence in Normed Vector Space has Unique Limit
- 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
- Cover of Interval By Closed Intervals is not Pairwise Disjoint
- Crossbar Theorem
- Cycle does not Contain Subcycles

### D

- Decomposition into Product of Power of 2 and Odd Integer is Unique
- Dedekind's Theorem
- Dedekind's Theorem/Proof 2
- Dedekind's Theorem/Proof 3
- Degree of Vertex/Examples/Impossible Order 4 Graph
- Dense-in-itself Subset of T1 Space is Infinite
- Derivative of Exponential Function/Proof 5/Lemma
- Derivative Operator on Continuously Differentiable Function Space with Supremum Norm is not Continuous
- Deterministic Time Hierarchy Theorem
- Diameter of Closure of Subset is Diameter of Subset
- Difference of Consecutive terms of Coherent Sequence
- 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/Existence
- Division Theorem/Positive Divisor/Positive Dividend/Existence/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
- Divisor of One of Coprime Numbers is Coprime to Other
- Divisor of One of Coprime Numbers is Coprime to Other/Proof 1
- Divisors of Power of Prime
- Domain of Continuous Injection to Hausdorff Space is Hausdorff
- Double Pointed Topology is not T0
- Double Pointed Topology is not T0/Proof 2
- Dual of Well-Ordering is not necessarily Well-Ordering

### E

- Edgeless Graph of Order n has n Components
- Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma
- Element of Matroid Base and Circuit has a Substitute/Lemma 2
- Element of Principal Ideal Domain is Finite Product of Irreducible Elements
- Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 2
- Empty Set as Subset
- Empty Set from Principle of Non-Contradiction
- Equal Alternate Angles implies Parallel Lines
- Equal Surfaces do not Intersect
- Equality of Natural Numbers
- Equality of Ordered Pairs/Necessary Condition
- Equality of Ordered Pairs/Necessary Condition/Proof from Empty Set Formalization
- Equality of Ordered Pairs/Necessary Condition/Proof from Wiener Formalization
- Equivalence of Definitions of Antisymmetric Relation
- Equivalence of Definitions of Arborescence
- Equivalence of Definitions of Ceiling Function
- Equivalence of Definitions of Closed Set in Normed Vector Space
- Equivalence of Definitions of Compact Topological Space
- Equivalence of Definitions of Connected Topological Space
- Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets
- 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 Everywhere Dense
- Equivalence of Definitions of Floor Function
- Equivalence of Definitions of Infimum of Real-Valued Function
- Equivalence of Definitions of Infinite Cyclic Group