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In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy.There is a distinction between a simple mistake 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 there is some element of concealment or ...
Showing that if the statement holds for an arbitrary number n ≥ b, then the same statement also holds for n + 1. This can be used, for example, to show that 2 n ≥ n + 5 for n ≥ 3. In this way, one can prove that some statement P(n) holds for all n ≥ 1, or even for all n ≥ −5.
convergence of the geometric series with first term 1 and ratio 1/2; Integer partition; Irrational number. irrationality of log 2 3; irrationality of the square root of 2; Mathematical induction. sum identity; Power rule. differential of x n; Product and Quotient Rules; Derivation of Product and Quotient rules for differentiating. Prime number
Contraposition is a logical inference that goes from a conditional statement to its contrapositive, which has its antecedent and consequent inverted and flipped. Learn the formal definition, the law of contraposition, the proof methods and the intuitive explanation of contraposition with examples.
Proof by exhaustion can be used to prove that if an integer is a perfect cube, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9. [3] Proof: Each perfect cube is the cube of some integer n, where n is either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these ...
Learn about the transformation rules that relate conjunctions and disjunctions via negation in propositional logic and Boolean algebra. See examples, applications, history and generalizations of De Morgan's laws.
Proof by example is a logical fallacy that uses one or more cases to illustrate a statement, rather than a full-fledged proof. Learn about the structure, argument form, and formal form of this fallacy, as well as some valid cases of proof by example in logic and mathematics.
Principia Mathematica is a three-volume work on the foundations of mathematics by Whitehead and Russell, published in 1910-1913. It aims to analyze, express, and solve the paradoxes of logic and set theory using symbolic logic and the theory of types.