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Form of mathematical proof
Mathematical induction is a method for proving that a statement P ( n ) {\displaystyle P(n)} is true for every natural number n {\displaystyle n} , that
Mathematical_induction
Method of logical reasoning
some degree of probability. Unlike deductive reasoning (such as mathematical induction), where the conclusion is certain, given the premises are correct
Inductive_reasoning
Reasoning for mathematical statements
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The
Mathematical_proof
Mathematical concept
Transfinite induction is an extension of mathematical induction to ordinal numbers. Its correctness is a theorem of ZF, and relies on the fact that the
Transfinite_induction
Process of repeating items in a self-similar way
function – Result of repeatedly applying a mathematical function Mathematical induction – Form of mathematical proof Mise en abyme – Technique of placing
Recursion
Subfield of automated reasoning and mathematical logic
reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major
Automated_theorem_proving
Collection of mathematical objects
In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects:
Set_(mathematics)
Proof method in mathematical logic
computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further
Structural_induction
Axioms for the natural numbers
ninth, final, axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to
Peano_axioms
Statement that all non empty subsets of positive numbers contains a least element
axioms for N {\displaystyle \mathbb {N} } , the induction axiom (or principle of mathematical induction), is logically equivalent to the well-ordering
Well-ordering_principle
Topics referred to by the same term
Word-sense induction Backward induction in game theory and economics Induced representation, in representation theory Mathematical induction, a method
Induction
Conformity to reality
establish theorems, such as direct proof, proof by contradiction, and mathematical induction. Formal logic studies the nature of deductive reasoning and the
Truth
Paradox arising from an incorrect proof
are the same color is a paradox that arises from a flawed use of mathematical induction to prove the statement All horses are the same color. There is no
All_horses_are_the_same_color
may be converted to for (i = 0; i < 10; ++i) { j = 1 << (i+1); } Mathematical induction Steven Muchnick; Muchnick and Associates (15 August 1997). Advanced
Induction_variable
Subfield of mathematics
(also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their
Mathematical_logic
Proof method in mathematical logic
of concurrent interacting objects. Coinduction is the mathematical dual to structural induction.[citation needed] Coinductively defined data types are
Coinduction
Production of voltage by a varying magnetic field
credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction. Lenz's law describes the
Electromagnetic_induction
Generalization of "n-th" to infinite cases
described here. In a broader mathematical sense, counting can be viewed as the instantiation of mathematical induction. To enumerate a well-ordered set
Ordinal_number
Type of mathematical proof
by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which a statement is established by
Proof_by_exhaustion
Defining elements of a set in terms of other elements in the set
programming Mathematical induction Recursive data types Recursion Recursion (computer science) Structural induction Henkin, Leon (1960). "On Mathematical Induction"
Recursive_definition
good proof." Polya begins Volume I with a discussion on induction, not mathematical induction, but as a way of guessing new results. He shows how the
Mathematics and Plausible Reasoning
Mathematics_and_Plausible_Reasoning
Certain type of mathematics from secondary school onwards
'O' Level Mathematics and Additional Mathematics, depending on the school. Some topics covered in this course include mathematical induction, complex number
Further_Mathematics
Algorithms which recursively solve subproblems
efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable
Divide-and-conquer_algorithm
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
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
Form of logic that allows quantification over predicates
2001. *Mendelson, Elliot (2009). Introduction to Mathematical Logic (hardcover). Discrete Mathematics and Its Applications (5th ed.). Boca Raton: Chapman
Second-order_logic
Basic law of electromagnetism
from electromagnetic induction (elaborated upon in the examples below). The laws of induction of electric currents in mathematical form were established
Faraday's_law_of_induction
Peano Mathematical induction Structural induction Recursive definition Naive set theory Element (mathematics) Ur-element Singleton (mathematics) Simple
List of mathematical logic topics
List_of_mathematical_logic_topics
Randomized algorithm
all inputs, thus providing the basis for a proof by mathematical induction. Here, the induction hypothesis is that the probability that a particular
Reservoir_sampling
some proofs Gödel's completeness theorem and its original proof Mathematical induction and a proof Proof that 0.999... equals 1 Proof that 22/7 exceeds
List_of_mathematical_proofs
Topics referred to by the same term
Natural induction can refer to: Mathematical induction Natural induction (labor) This disambiguation page lists articles associated with the title Natural
Natural_induction
Theorem: (cos x + i sin x)^n = cos 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. For
De_Moivre's_formula
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern
History_of_mathematics
American scientist (1839–1914)
concept of abductive reasoning, as well as rigorously formulating mathematical induction and deductive reasoning. He was one of the founders of statistics
Charles_Sanders_Peirce
Topics referred to by the same term
can inherit behaviors and features from more than one superclass Mathematical induction Mutual information, a measure of mutual dependence of two random
MI
Algebraic expansion of powers of a binomial
Armstrong in Rashed, Roshdi (1994). "Mathematical Induction: al-Karajī and al-Samawʾal". The Development of Arabic Mathematics: Between Arithmetic and Algebra
Binomial_theorem
step. mathematical induction schema Synonym of mathematical induction. mathematical logic The study of logic within the framework of mathematical reasoning
Glossary_of_logic
Italian mathematician and glottologist (1858–1932)
systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin. He also created
Giuseppe_Peano
Axiomatic logical system
1950. It is usually denoted Q. Q is PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories
Robinson_arithmetic
Sequence of operations for a task
program is that it lends itself to proofs of correctness using mathematical induction. By themselves, algorithms are not usually patentable. In the United
Algorithm
Arithmetic mean is greater than or equal to geometric mean
apply mathematical induction and only well-known rules of arithmetic. Induction basis: For n = 1 the statement is true with equality. Induction hypothesis:
AM–GM_inequality
American mathematician
Henkin, L. (1960). On mathematical induction. The American Mathematical Monthly. 67(4), 323-338. Henkin, L. (1961). Mathematical Induction. En MAA Film Manual
Leon_Henkin
Type of binary relation
the graph of the successor function x ↦ x+1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion. If
Well-founded_relation
influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains
Mathematics in the medieval Islamic world
Mathematics_in_the_medieval_Islamic_world
following list features abbreviated names of mathematical functions, function-like operators and other mathematical terminology. This list is limited to abbreviations
List of mathematical abbreviations
List_of_mathematical_abbreviations
Cumulative effect produced when one event sets off a chain of other events
syndrome – Theoretical satellite collision cascade Mathematical induction – Form of mathematical proof Placebo effect – Substance or treatment of no
Domino_effect
Algorithm for finding shortest paths
denser graphs. To prove the correctness of Dijkstra's algorithm, mathematical induction can be used on the number of visited nodes. Invariant hypothesis:
Dijkstra's_algorithm
American columnist, author and lecturer (born 1946)
other mathematical problems. Reviewers questioned her criticism of Wiles' proof, asking whether it was based on a correct understanding of mathematical induction
Marilyn_vos_Savant
intuitionistic type theory (ITT), a discipline within mathematical logic, induction-induction is for simultaneously declaring some inductive type and
Induction-induction
Subset of artificial intelligence
here refers to philosophical induction, suggesting a theory to explain observed facts, rather than mathematical induction, proving a property for all members
Machine_learning
Topics referred to by the same term
recursive formula for a sequence of numbers a n {\displaystyle a_{n}} Mathematical induction, a method of proof also called "proof by recursion" Recursion, a
Recursion_(disambiguation)
Latin phrase meaning 'continuing forever'
Siphonaptera. Look up ad infinitum in Wiktionary, the free dictionary. Mathematical induction Recursion Self-reference "The Song That Never Ends" Turtles all
Ad_infinitum
In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron
of an n-dimensional hypercube's diagonals can be calculated by mathematical induction. The longest diagonal of an n-cube is n {\displaystyle {\sqrt {n}}}
Diagonal
Summation formula
Euler–Maclaurin summation formula which can be formalized by mathematical induction, in which the induction step relies on integration by parts and on identities
Euler–Maclaurin_formula
Method of deriving conclusions
proof by contradiction, and mathematical induction. Mathematical logic, a subfield of mathematics and logic, uses mathematical methods and frameworks to
Rule_of_inference
Axiomatic set theories based on the principles of mathematical constructivism
the previous induction principles, one has full set induction, which is to be compared to well-founded induction. Like mathematical induction above, the
Constructive_set_theory
British mathematician and logician (1806–1871)
conjunction, disjunction, and negation, and for coining the term "mathematical induction", the underlying principles of which he formalized. De Morgan's
Augustus_De_Morgan
Mathematical theorem
} We use mathematical induction. For n = 0 this is just the assumed integral inequality, because the empty sum is defined as zero. Induction step from
Grönwall's_inequality
Cyclic algorithm to solve indeterminate quadratic equations
marvellous height of mathematical complexity. This method is also known as the cyclic method and contains traces of mathematical induction. Chakra in Sanskrit
Chakravala_method
Figurate number
{4(4+1)}{2}}=10} (green). This formula can be proven formally using mathematical induction. It is clearly true for 1 {\displaystyle 1} : T 1 = ∑ k = 1 1 k
Triangular_number
Branch of mathematical logic
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining
Reverse_mathematics
American illusionists and entertainers
generalized it so it would work with any number of initial cards, using mathematical induction; their complete analysis was published in the Journal of Magic Research
Penn_&_Teller
Concept in the philosophy of mathematics
This type of process occurs in mathematics, for instance, in standard formalizations of the notions of mathematical induction, infinite series, infinite products
Actual_and_potential_infinity
geomorphology and natural climate change. 1000: Al-Karaji uses mathematical induction. 1058: al-Zarqālī in Islamic Spain discovers the apsidal precession
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Mathematical puzzle game
rigorous mathematical proof with mathematical induction and is often used as an example of recursion when teaching programming. As in many mathematical puzzles
Tower_of_Hanoi
Triangular array of the binomial coefficients
triangle. It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem. Since ( a + b ) n = b n ( a b + 1 ) n
Pascal's_triangle
Numbers obtained by adding the two previous ones
comparing areas: Fibonacci identities often can be easily proved using mathematical induction. For example, reconsider ∑ i = 1 n F i = F n + 2 − 1. {\displaystyle
Fibonacci_sequence
Question of whether inductive reasoning leads to definitive knowledge
The problem of induction is a philosophical problem that questions the rationality of predictions about unobserved things based on previous observations
Problem_of_induction
Mathematical relationships
There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's
QM–AM–GM–HM_inequalities
Certain type of mistaken proof
and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies
Mathematical_fallacy
Type of AC electric motor
electromagnetic induction from the magnetic field of the stator winding. An induction motor therefore needs no electrical connections to the rotor. An induction motor's
Induction_motor
Ability to readily identify logical or mathematical truth
logical or mathematical truth—and the ability to solve mathematical challenges efficiently. Humans apply logical intuition in proving mathematical theorems
Logical_intuition
Every graph has evenly many odd vertices
number of odd-degree vertices. Alternatively, it is possible to use mathematical induction to prove the degree sum formula, or to prove directly that the number
Handshaking_lemma
Foundational controversy in twentieth-century mathematics
types of mathematical induction: (1) the formal induction rule (Peano's axiom); (2) the inductive definition (examples: counting, "proof by induction"); and
Brouwer–Hilbert_controversy
Smallest example which falsifies a claim
by the most usual formulation of mathematical induction; but the scope of the method can include well-ordered induction of any kind. The minimal counterexample
Minimal_counterexample
Mathematically obvious
sake of completeness cannot be ignored. For instance, proofs by mathematical induction have two parts: the "base case" which shows that the theorem is
Triviality_(mathematics)
Inequality about exponentiations of ''1+x''
non-negative integer and x ≥ − 2 {\displaystyle x\geq -2} , using mathematical induction in the following form: we prove the inequality for r ∈ { 0 , 1 }
Bernoulli's_inequality
Undergraduate admission exam of China
Fundamental Theorem of Calculus, Simple Application of Definite Integral, Mathematical Induction, Counting Principle, Random Variable and Its Distribution. For candidates
Gaokao
17th-century conjecture proved by Andrew Wiles in 1994
condition), then θ must divide the product xyz. Her goal was to use mathematical induction to prove that, for any given p, infinitely many auxiliary primes
Fermat's_Last_Theorem
Theorem in real analysis
The proof uses mathematical induction. The case n = 1 is simply the standard version of Rolle's theorem. For n > 1, take as the induction hypothesis that
Rolle's_theorem
Geometric shape formed from three squares
board can be completely covered with L-trominoes. To prove this by mathematical induction, partition the board into a quarter-board of size 2n−1 × 2n−1 that
Tromino
Medieval Jewish philosopher
combinatorial identities. The work is notable for its early use of proof by mathematical induction, and pioneering work in combinatorics. The title Maaseh Hoshev literally
Gersonides
Supplementary pair of angles at each vertex of a polygon
where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180°, then
Internal_and_external_angles
Recurrence relations of binomial coefficients in Pascal's triangle
k}={n-1 \choose k-1}+{n-1 \choose k}.} This identity can be proven by mathematical induction on n {\displaystyle n} . Base case Let n = r {\displaystyle n=r}
Hockey-stick_identity
Properties of mathematical relationships
f ( x ) {\displaystyle f(nx)=nf(x)} for any natural number n by mathematical induction, and then n f ( x ) = f ( n x ) = f ( m n m x ) = m f ( n m x )
Linearity
Extension of modal logic
often we perform a {\displaystyle a\,\!} . A6 is recognizable as mathematical induction with the action n := n+1 of incrementing n generalized to arbitrary
Dynamic_logic_(modal_logic)
Generalization of the product rule in calculus
f^{(0)}=f} ). The rule can be proven by using the product rule and mathematical induction. If, for example, n = 2, the rule gives an expression for the second
General_Leibniz_rule
Generalization of the real numbers
This group of definitions is recursive, and requires some form of mathematical induction to define the universe of objects (forms and numbers) that occur
Surreal_number
Conditional statement which is true because the antecedent cannot be satisfied
frequently arise as the base case of proofs by mathematical induction. This notion has relevance in pure mathematics, as well as in any other field that uses
Vacuous_truth
Number of subsets of a given size
the important recurrence relation which can be used to prove by mathematical induction that ( n k ) {\displaystyle {\tbinom {n}{k}}} is a natural number
Binomial_coefficient
German mathematician (1862–1943)
developed important tools used in modern mathematical physics. He was a co-founder of proof theory and mathematical logic. Hilbert, the first of two children
David_Hilbert
Arithmetic operation
the associative and commutative properties, among others, through mathematical induction. The simplest conception of an integer is that it consists of an
Addition
Axiomatic set theory devised by W.V.O. Quine
the intersection of all inductive sets. This definition enables mathematical induction for stratified statements P ( n ) {\displaystyle P(n)} , because
New_Foundations
"Chapter 4, eqn 4.3.45". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint
List of trigonometric identities
List_of_trigonometric_identities
Mathematical proofs of basic properties of addition of the natural numbers
This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity
Proofs involving the addition of natural numbers
Proofs_involving_the_addition_of_natural_numbers
Topics referred to by the same term
not use recursion to produce an answer Base case (induction), the basis in mathematical induction, showing that a statement holds for the lowest possible
Base_case
competition, such as the USA Mathematical Olympiad, or the International Mathematical Olympiad. Students majoring in mathematics, the physical sciences, engineering
Mathematics education in the United States
Mathematics_education_in_the_United_States
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
French polymath (1623–1662)
treatise, Pascal gave an explicit statement of the principle of mathematical induction. In 1654, he proved Pascal's identity relating the sums of the p-th
Blaise_Pascal
Persian mathematician and engineer (c. 953 – c. 1029)
quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction. As Katz says Another important idea introduced by al-Karaji and
Al-Karaji
MATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION
Girl/Female
Gujarati, Hindu, Indian, Kannada, Telugu
Mathematician
Boy/Male
Australian, Vietnamese
Complete; Mathematics
Boy/Male
Bengali, Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
One who Calculates; Astrologer; Mathematician
Surname or Lastname
English
English : habitational name from a place in West Yorkshire named Colden, from Old English cald ‘cold’ col ‘charcoal’ + denu ‘valley’.English and Scottish : variant of Cowden.Cadwallader Colden (1688–1778), physician, botanist, and mathematician, who for fifteen years was lieutenant-governor of New York colony, was born in Dalkeith, Scotland.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Telugu
An Astrologer; Mathematician
Girl/Female
Hindu
Mathematician
Girl/Female
Tamil
Mathematician
MATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION
Boy/Male
Hindu
All
Girl/Female
English Celtic Welsh
Friend.
Female
Greek
(Διώνη) Greek name DIÔNÊ means "the goddess." In mythology, this is the name of the Titan mother of Aphrodite. It is a feminine form of Zeus.Â
Boy/Male
Indian, Sanskrit
Lord of Perfection
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sikh, Tamil, Telugu
A Tree
Girl/Female
Arabic, Australian, Muslim
Rain Drops
Male
Greek
Greek name PYROIS means "fiery." In mythology, this is the name of one of the horses of the Sun.
Girl/Female
Arabic, Australian, Muslim
Shy
Boy/Male
Arabic, Muslim
Happy
Girl/Female
American, Australian, Dutch, Finnish, French, German, Hebrew, Swedish
Bitterness; Royal Lady; Similar to Maria; From the God Mars; Of the Sea
MATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION
a.
Producing mathematically perfect harmony or concord; sweetly or perfectly harmonious.
n.
One skilled in geometry; a geometer; a mathematician.
a.
Of or pertaining to mathematical calculations; performing or able to perform mathematical calculations.
n.
Mixed mathematics.
v. i.
To alter or change in succession; to alternate; as, one mathematical quantity varies inversely as another.
a.
See Mathematical.
n.
Any lineal or mathematical diagram; an outline.
n.
Learning; especially, mathematics.
a.
Of or pertaining to mathematics; according to mathematics; hence, theoretically precise; accurate; as, mathematical geography; mathematical instruments; mathematical exactness.
n.
The act or process of making mathematical computations or of estimating results.
n.
That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.
n.
One skilled in geometry; a geometrician; a mathematician.
v. i.
To use figures in a mathematical process; to do sums in arithmetic.
n.
A solution, the result of a mathematical operation; as, the answer to a problem.
n.
The symbol, quantity, or thing upon which a mathematical operation is performed; -- called also faciend.
a.
Pertaining to, or having the nature of, an anathema.
a.
Pertaining to Euler, a German mathematician of the 18th century.
a.
Alt. of Anathematical
v.
A mathematical point; -- regularly used in old English translations of Euclid.
n.
One versed in mathematics.