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Form of mathematical proof
the next one (the step). — Concrete Mathematics, page 3 margins. A proof by induction consists of two cases. The first, the base case, proves the statement
Mathematical_induction
Science fiction short story
"Proof by Induction" is a 2021 science fiction short story by José Pablo Iriarte. It was first published in Uncanny Magazine. Paulie is a mathematician
Proof_by_Induction
Type of mathematical proof
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof
Proof_by_exhaustion
Arithmetic mean is greater than or equal to geometric mean
following proof by cases relies directly on well-known rules of arithmetic but employs the rarely used technique of forward-backward-induction. It is essentially
AM–GM_inequality
Method of logical reasoning
procedure like proof by exhaustion. Both mathematical induction and proof by exhaustion are examples of complete induction. Complete induction is a masked
Inductive_reasoning
Reasoning for mathematical statements
phrase "proof by induction" is often used instead of "proof by mathematical induction". Proof by contraposition infers the statement "if p then q" by establishing
Mathematical_proof
Proof method in mathematical logic
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some
Structural_induction
Election result probability theorem
=1-2{\frac {q}{p+q}}={\frac {p-q}{p+q}}} Another method of proof is by mathematical induction: We loosen the condition p > q {\displaystyle p>q} to p ≥
Bertrand's_ballot_theorem
Paradox arising from an incorrect proof
not exist, and he had an infinite number of limbs. The argument is proof by induction. First, we establish a base case for one horse ( n = 1 {\displaystyle
All_horses_are_the_same_color
Mathematical identity
{-b_{2}}{1+b_{2}+}}\cdots {\frac {-b_{n}}{1+b_{n}}}\neq -1.} Proof: We perform a double induction. For n = 1 {\displaystyle n=1} , we have a 0 1 + − a 1 1
Euler's continued fraction formula
Euler's_continued_fraction_formula
Topics referred to by the same term
induction Backward induction in game theory and economics Induced representation, in representation theory Mathematical induction, a method of proof Strong
Induction
Mathematical identities for the Fibonacci numbers
1960 by Dustan Everman as problem 1396 in The American Mathematical Monthly, and in 1901 by Alberto Tagiuri in Periodico di Matematica. A quick proof of
Cassini and Catalan identities
Cassini_and_Catalan_identities
1995 publication in mathematics
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Theorem on triangulation graph colorings
Rabee Tourky presented a different proof, using the volume of a simplex. It proceeds in one step, with no induction. Suppose there is a d-dimensional simplex
Sperner's_lemma
On prime factors of integer products
in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains. Euclid's lemma is commonly
Euclid's_lemma
Mathematical proofs of basic properties of addition of the natural numbers
we have a + 1 = 1 + a. We will prove this by induction on a (an induction proof within an induction proof). We have proved that 0 commutes with everything
Proofs involving the addition of natural numbers
Proofs_involving_the_addition_of_natural_numbers
Certain type of mistaken proof
fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing
Mathematical_fallacy
Inequality applying to probability spaces
replaced by any measure. Boole's inequality may be proved for finite collections of n {\displaystyle n} events using the method of induction.[citation
Boole's_inequality
Theorem in mathematics
above, it suffices to treat the upper bound in (1). For a proof by mathematical induction, we start with n = 2. {\displaystyle n=2.} Observe that x 1
Rearrangement_inequality
Theorem: (cos x + i sin x)^n = cos nx + i sin nx
nx+i\sin nx.} The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there
De_Moivre's_formula
Method of differentiating single-term polynomials
kx^{k-1}=x^{k}+kx^{k}=(k+1)x^{k}=(k+1)x^{(k+1)-1}} By the principle of mathematical induction, the statement is true for all natural numbers n. Let
Power_rule
17th-century conjecture proved by Andrew Wiles in 1994
efforts by releasing prior work in small segments as separate papers and confiding only in his wife. His initial study suggested proof by induction, and
Fermat's_Last_Theorem
Mathematical concept
existence of which is guaranteed by the fact the class of ordinal numbers is well-ordered. A proof by transfinite induction is often broken down into three
Transfinite_induction
Way of arriving to a mathematical proof
including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction. A direct proof is the simplest form of proof there
Direct_proof
Smallest example which falsifies a claim
the methods of proof by induction and proof by contradiction. More specifically, in trying to prove a proposition P, one first assumes by contradiction
Minimal_counterexample
Foundational controversy in twentieth-century mathematics
mathematical induction: (1) the formal induction rule (Peano's axiom); (2) the inductive definition (examples: counting, "proof by induction"); and (3)
Brouwer–Hilbert_controversy
Low-space search for a majority element
pairs of unequal elements, and c copies of m left over. This is a proof by induction; it is trivially true when n = c = 0, and is maintained every time
Boyer–Moore majority vote algorithm
Boyer–Moore_majority_vote_algorithm
Mathematical logic concept
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of
Gentzen's_consistency_proof
Statement that all non empty subsets of positive numbers contains a least element
the equivalence between induction and well-ordering is a common result, Lars-Daniel Öhman has argued that "proofs" of induction based on well-ordering
Well-ordering_principle
Set that is not a finite set
sets to start to explain infinite sets using proof concepts such as mapping, proof by induction, or proof by contradiction. Mathematical trees can also
Infinite_set
Theorem in the mathematics of Lie's theory
algebra. A more geometric proof is due to Élie Cartan and was published by Willem van Est [nl]. This proof uses induction on the dimension of the center
Lie's_third_theorem
On the smallest non-interesting number
interesting fact concerning each of the positive integers. Here is a "proof by induction" that such is the case. Certainly, 1, which is a factor of each positive
Interesting_number_paradox
Type of AC electric motor
An induction motor or asynchronous motor is an AC electric motor in which the electric current in the rotor that produces torque is obtained by electromagnetic
Induction_motor
Mathematical model of the physical space
formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g.,
Euclidean_geometry
its original proof Mathematical induction and a proof Proof that 0.999... equals 1 Proof that 22/7 exceeds π Proof that e is irrational Proof that π is irrational
List_of_mathematical_proofs
Result concerning ideals of commutative rings
is not contained in the union ⋃ I i {\textstyle \bigcup I_{i}} . Proof by induction on n: The idea is to find an element of R that is in E and not in
Prime_avoidance_lemma
Basic law of electromagnetism
of induction describes how a changing magnetic field can induce an electric current in a circuit. This phenomenon, known as electromagnetic induction, is
Faraday's_law_of_induction
Axioms for the natural numbers
theory. In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. Gentzen explained:
Peano_axioms
Card game
determine the best strategy that the other player should use. Using a proof by induction on the number of cards, Ross showed that the optimal strategy for
Goofspiel
Matrix that, squared, equals itself
integers n, A n = A {\displaystyle A^{n}=A} . This can be shown using proof by induction. Clearly we have the result for n = 1 {\displaystyle n=1} , as A 1
Idempotent_matrix
Limited form of tree data structure
at the Wayback Machine entry in the FindStat database Binary Tree Proof by Induction Archived 2019-04-07 at the Wayback Machine Balanced binary search
Binary_tree
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an
Original proof of Gödel's completeness theorem
Original_proof_of_Gödel's_completeness_theorem
On chains and antichains in partial orders
common size of the antichain and chain decomposition. The following proof by induction on the size of the partially ordered set P {\displaystyle P} is based
Dilworth's_theorem
2001 textbook by Graham Priest
brief "mathematical prolegomenon" covering set-theoretic notation and proof by induction, advanced material on metatheory would still require graduate-level
An Introduction to Non-Classical Logic
An_Introduction_to_Non-Classical_Logic
Concept in mathematical logic
to the function. Induction-recursion can be used to define large types including various universe constructions. It increases the proof-theoretic strength
Induction-recursion
Subfield of automated reasoning and mathematical logic
Retrieved 2022-11-20. Bundy, Alan (1999). The automation of proof by mathematical induction (PDF) (Technical report). Informatics Research Report. Vol
Automated_theorem_proving
Mathematical technique used in proof theory
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories
Ordinal_analysis
Kind of transfinite induction
In set theory, ∈ {\displaystyle \in } -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets
Epsilon-induction
Concept in logic
definition of formulas in ZFC. The definition is recursive, so a proof by induction is used. In ZFC in first-order logic without equality, "set equality"
Substitution_(logic)
Construction in homological algebra
of R / ( x 1 , … , x r ) {\displaystyle R/(x_{1},\dots ,x_{r})} . Proof by induction on r. If r = 1 {\displaystyle r=1} , then H 1 ( K ( x 1 ; M ) )
Koszul_complex
Formal power series in algebra
coefficients divided by ∏ ( 1 − t d i ) {\displaystyle \prod (1-t^{d_{i}})} . The standard proof today is an induction on n. Hilbert's original proof made a use
Hilbert–Poincaré_series
Conclusion made on the basis of one or few instances of a phenomenon
basis of one or a few instances of that phenomenon. It is similar to a proof by example in mathematics. It is an example of jumping to conclusions. For
Faulty_generalization
Doubly exponential integer sequence
Integer Sequences. OEIS Foundation. Nešetřil & Matoušek (1998). A proof by induction is given by Sylvester (1880), p. 333. Graham, Knuth & Patashnik (1989),
Sylvester's_sequence
Speculative Fiction Writing Circle
José Pablo Iriarte, author of Nebula Award-nominated short story "Proof by Induction" and Nebula Award- and James Tiptree Award-nominated novelette "The
Codex_Writers_Group
Concept in distributed computing
execution of A in R, all the processes have the same states. Proof. Proof by induction on k {\displaystyle k} . Base case: k = 0 {\displaystyle k=0}
Leader_election
principle of induction was given by Pascal in his Traité du triangle arithmétique (1665).[citation needed] In between, implicit proof by induction for arithmetic
Mathematics in the medieval Islamic world
Mathematics_in_the_medieval_Islamic_world
Deleting a graph edge and merging its nodes
G} . Both edge and vertex contraction techniques are valuable in proof by induction on the number of vertices or edges in a graph, where it can be assumed
Edge_contraction
Properties linking logical conjunction and disjunction
have that φ = ¬ ( φ ∗ ) {\textstyle \varphi =\neg (\varphi ^{*})} . Proof: By induction on complexity. For the base case, we consider an arbitrary atomic
Conjunction/disjunction duality
Conjunction/disjunction_duality
Infinite graph containing all countable graphs
partial copy, with one more vertex. This method forms the basis for a proof by induction, with the 0-vertex subgraph as its base case, that every finite or
Rado_graph
Planar graphs have straight drawings
vertices of P do not cross any other edges, completing the proof. The induction step of this proof is illustrated at right. De Fraysseix, Pach and Pollack
Fáry's_theorem
Erroneous method of proof
full-fledged proof. Affirming the consequent Anecdotal evidence Bayesian probability Counterexample Hand-waving Inductive reasoning Problem of induction Modus
Proof_by_example
Functional programming language
This way of writing recursive functions/inductive proofs is more natural than applying raw induction principles. In Agda, dependently typed pattern matching
Agda_(programming_language)
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
underlying vector space has a basis composed of Jordan chains. We give a proof by induction that any complex-valued square matrix A may be put in Jordan normal
Jordan_normal_form
Inference seeking the simplest and most likely explanation
knowledge is one matted felt of pure hypothesis confirmed and refined by induction. Not the smallest advance can be made in knowledge beyond the stage of
Abductive_reasoning
Branch of mathematical logic
formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures
Proof_theory
Use of functions that call themselves
n=0} . The recursive case is analogous to the inductive step in a proof by induction: it assumes that the function works for a smaller instance and then
Recursion_(computer_science)
Argument in the philosophy of science
In the philosophy of science, the pessimistic induction, also known as the pessimistic meta-induction, is an argument that seeks to rebut scientific realism
Pessimistic_induction
mathematicians). Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction. In 1882
Proof_that_pi_is_irrational
Theorem in mathematical analysis
{\displaystyle u\in L^{r}(\mathbb {R} ^{n})} . The core of the proof is based on two proofs by induction. In many problems coming from the theory of partial differential
Gagliardo–Nirenberg interpolation inequality
Gagliardo–Nirenberg_interpolation_inequality
Generalization of "n-th" to infinite cases
of proof by infinite descent, which can be generalized to ordinals and other well-ordered classes too, as a special case of transfinite induction where
Ordinal_number
Form of reasoning
often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is top-down while induction is bottom-up
Deductive_reasoning
freshman's dream. Leaving the proof for later on, we proceed with the induction. Proof. Assume kp ≡ k (mod p), and consider (k+1)p. By the lemma we have ( k +
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Annual awards for science fiction or fantasy
ballot was announced. Bellet's story was replaced on the ballot by "A Single Samurai" by Steven Diamond. Jordison, Sam (August 7, 2008). "An International
Hugo Award for Best Short Story
Hugo_Award_for_Best_Short_Story
Non-contradiction of a theory
Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann
Consistency
Branch of mathematical logic
full second-order induction scheme either. This restriction is important: systems with restricted induction have significantly lower proof-theoretical ordinals
Reverse_mathematics
essentially dissimilar concepts as though they were essentially similar. Proof by assertion – a proposition is repeatedly restated regardless of contradiction;
List_of_fallacies
Logical fallacy in which the conclusion provides the premise
defect in an argument whereby the premises are just as much in need of proof or evidence as the conclusion. As a consequence, the argument becomes a
Circular_reasoning
American author
"This Wine-Dark Feeling That Isn't the Blues" (Escape Pod, 2019) "Proof by Induction" (Uncanny Magazine, 2021) Benny Ramírez and the Nearly Departed (Knopf
José_Pablo_Iriarte
Theorem relating Milnor K-theory and Galois cohomology
part of the Beilinson-Lichtenbaum conjectures. It often occurs in proofs by induction that the statement being proved has to be strengthened in order to
Norm residue isomorphism theorem
Norm_residue_isomorphism_theorem
Graph-theoretic description of polyhedra
convex polyhedron. There are three standard approaches for this part: proofs by induction, lifting two-dimensional Tutte embeddings into three dimensions using
Steinitz's_theorem
1748 book by David Hume
belief by association. Habit makes us expect the past to repeat; this expectation produces belief, not a rational proof. Hence the “problem of induction”:
An Enquiry Concerning Human Understanding
An_Enquiry_Concerning_Human_Understanding
Problem in probability theory
known as Pascal's triangle (including several of the first explicit proofs by induction) Pascal finally showed that in a game where player a needs r points
Problem_of_points
simply "given", it is natural to see the principles of proof by mathematical induction and definition by recursion along that system as "given" as well. .
John_Penn_Mayberry
Foundational result in symplectic geometry
+x_{p}\,\mathrm {d} y_{p}+\mathrm {d} x_{p+1}.} Darboux's original proof used induction on p {\displaystyle p} and it can be equivalently presented in terms
Darboux's_theorem
Methods in artificial intelligence research
the Mace4 model checker. ACL2 is a theorem prover that can handle proofs by induction and is a descendant of the Boyer-Moore Theorem Prover, also known
Symbolic artificial intelligence
Symbolic_artificial_intelligence
report). IRIA. p. 8. 283. Huet, G.; Hullot, J.M. (October 1980). "Proofs by Induction in Equational Theories with Constructors". 21st Annual Symposium
Gérard_Huet
Proof by Alan Turing
Turing's proof is a proof by Alan Turing submitted on 12 November 1936 and first published in 1937 with the title "On Computable Numbers, with an Application
Turing's_proof
It leads to Russell's paradox. basis The initial case in a proof by mathematical induction. Bayes' theorem A theorem in probability theory used to update
Glossary_of_logic
Fundamental result in the branch of mathematics known as character theory
subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations
Brauer's theorem on induced characters
Brauer's_theorem_on_induced_characters
Mathematical theory
uncomputable. The proof of this is derived from a game between the induction and the environment. Essentially, any computable induction can be tricked by a computable
Solomonoff's theory of inductive inference
Solomonoff's_theory_of_inductive_inference
Interpretation of intuitionistic logic
what is intended to be a proof of a given formula. This is specified by induction on the structure of that formula: A proof of P ∧ Q {\displaystyle P\wedge
Brouwer–Heyting–Kolmogorov interpretation
Brouwer–Heyting–Kolmogorov_interpretation
Description of non-logical symbols
However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be
Signature_(logic)
2025 anthology edited by Stephen Kotowych
Miller) "Mr. Death" [best short story nominee] (Alix E. Harrow) "Proof by Induction" [best short story nominee] (José Pablo Iriarte) "Laughter Among the
Nebula_Awards_Showcase_57
Heuristics in automated theorem proving
the proof. We aim to show that the addition of natural numbers is commutative. This is an elementary property, and the proof is by routine induction. Nevertheless
Rippling
Mathematical criterion in game theory
s_{1}} has been eliminated from the game, and complete the proof by induction. It may come by surprise then that weakly max-solvable games do not necessarily
Max-dominated_strategy
Topics referred to by the same term
{\displaystyle a_{n}} Mathematical induction, a method of proof also called "proof by recursion" Recursion, a 2004 science fiction novel by Tony Ballantyne Recursion
Recursion_(disambiguation)
Programming language
sequences, sets, multisets, infinite sequences and sets, induction, co-induction, and calculational proofs. Verification obligations are discharged automatically
Dafny
Proof in set theory
is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, but it was not his first proof of the uncountability
Cantor's_diagonal_argument
term has a unique normal form. Proof Because → {\displaystyle \to } is terminating, we can perform well-founded induction on u {\displaystyle u} along →
Newman's_lemma
Bar induction is a reasoning principle used in intuitionistic mathematics, introduced by L. E. J. Brouwer. Bar induction's main use is the intuitionistic
Bar_induction
PROOF BY-INDUCTION
PROOF BY-INDUCTION
Girl/Female
Indian
Witness; Proof
Girl/Female
Muslim
Guide, Proof
Boy/Male
Arabic, Muslim
The Proof
Boy/Male
Arabic, Muslim
Evidence; Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
Boy/Male
Muslim
Proof
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Indian
Proof
Boy/Male
Indian
Argument, Reasoning, Proof
Boy/Male
Muslim
Argument, Reasoning, Proof
Boy/Male
Arabic
Proof; Evidence
Boy/Male
Muslim
Proof
Boy/Male
Indian
Proof
Girl/Female
Muslim
Proof
Boy/Male
Arabic, French, German, Gujarati, Hindu, Indian, Malaysian, Muslim, Turkish
Proof; Evidence
Boy/Male
Muslim/Islamic
Proof
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Muslim
Evidence. Proof.
Boy/Male
Muslim/Islamic
Proof
Surname or Lastname
English
English : variant of Rolfe.German : from Ruffo, a short form of a personal name formed with hrÅd ‘renown’, ‘victory’.Probably an Americanized spelling of German Ruf and Ruff.
PROOF BY-INDUCTION
PROOF BY-INDUCTION
Boy/Male
Indian, Telugu
Son of Wind
Girl/Female
Tamil
Knowledge, Wisdom
Girl/Female
Hindu, Indian, Kannada
Sweet
Boy/Male
Indian, Sanskrit
Who Works According to Schedule; Organized; Planned
Boy/Male
Bengali, Indian
King of Earth
Girl/Female
Tamil
Abode, Existence
Girl/Female
Muslim/Islamic
Flower or fruit
Boy/Male
Muslim
Cold, Mild
Surname or Lastname
English and German
English and German : occupational name from Middle English, Middle Low German peller ‘maker (or seller) of expensive cloth’, derived from Old English pæll, pell ‘costly or purple cloth or cloak’, Middle Low German pelle (see Pelle 2).Southern English : topographic name for someone living by an inlet of the sea, a derivative of Old English pyll ‘inlet’ (see Pill 1) + the -er suffix denoting an inhabitant.German : from a Germanic personal name formed with bald ‘brave’ + heri ‘army’.
Girl/Female
Australian, Chinese, German
Flower Bud
PROOF BY-INDUCTION
PROOF BY-INDUCTION
PROOF BY-INDUCTION
PROOF BY-INDUCTION
PROOF BY-INDUCTION
v. t.
To cover with a roof.
v. t.
To arm with proof armor; to arm securely; as, to proof-arm herself.
n.
A trial impression, as from type, taken for correction or examination; -- called also proof sheet.
a.
Proof against proofs; obstinate in the wrong.
a.
Firm or successful in resisting; as, proof against harm; waterproof; bombproof.
adv.
Passing near; going past; past; beyond; as, the procession has gone by; a bird flew by.
n.
That which resembles, or corresponds to, the covering or the ceiling of a house; as, the roof of a cavern; the roof of the mouth.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
n.
Proof.
v. t.
Armor of excellent or tried quality, and deemed impenetrable; properly, armor of proof.
a.
Used in proving or testing; as, a proof load, or proof charge.
n.
Proof.
adv.
Aside; as, to lay by; to put by.
a.
Out of the common path; aside; -- used in composition, giving the meaning of something aside, secondary, or incidental, or collateral matter, a thing private or avoiding notice; as, by-line, by-place, by-play, by-street. It was formerly more freely used in composition than it is now; as, by-business, by-concernment, by-design, by-interest, etc.
a.
Proof against penetration or permeation by water; impervious to water; as, a waterproof garment; a waterproof roof.
n.
Proof.
n.
The cover of any building, including the roofing (see Roofing) and all the materials and construction necessary to carry and maintain the same upon the walls or other uprights. In the case of a building with vaulted ceilings protected by an outer roof, some writers call the vault the roof, and the outer protection the roof mask. It is better, however, to consider the vault as the ceiling only, in cases where it has farther covering.
pref.
With, as means, way, process, etc.; through means of; with aid of; through; through the act or agency of; as, a city is destroyed by fire; profit is made by commerce; to take by force.
n.
Proof.
n.
Proof by witness; attestation; testimony.