2. 反证法 (Proof by Contradiction) Example 3: Example 4: 附:无限递降法(不需要掌握) Example 5: 3. 寻找反例 (Proof by Counterexample / Counterclaim) Example 2: 4. 归纳法 (Proof by Induction) Example 6: Example 7: 总结 在数学AA课程大纲中,所有会考到的证明方法在课本中被归为四种: 直接证...
TWO EXAMPLES OF PROOF BY MATHEMATICAL INDUCTION.DR. LOMONACOProposition: Use the principle of mathematical induction to prove that P (n) :nΣj=1j2 = n(n + 1) (2n + 1) 6 , for all integers n ^ 1. Proof (by weak induction): Basis Step: P(n) is true for n = 1, for:1Σj...
Proof by mathematical induction An example of the application of mathematical induction in the simplest case is the proof that the sum of the first n odd positive integers is n2—that is, that (1.) 1 + 3 + 5 +⋯+ (2n − 1) = n2 for every positive integer n. Let F be the ...
One is Proof by Exhaustion, the other is Universal Generalisation/Introduction - or as it is called in the book - Generalising from the Generic Particular. 【Week 4 Mathematical Induction and Recursion】 The Principle of Mathematical Induction is as follows: ▶ Let Pn be defined for integers ...
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers:[15] Let N = {1,2,3,4,...} be the set of natural numbers, and P(n) be a mathematical statement involving the natural number n belonging to...
Mathematical induction is required for reasoning about objects or events containing repetition, e.g. computer programs with recursion or iteration, electronic circuits with feedback loops or parameterized components, and properties that hold for all time forward. It is thus a vital ingredient of formal...
mathematical induction (redirected fromMathematical induction Proof) Encyclopedia n. Induction. American Heritage® Dictionary of the English Language, Fifth Edition. Copyright © 2016 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reser...
Modus Ponens Universal Instantiation Example: "All men are mortal, Socrates is a man, therefore Socrates is a mortal." x A(x) => A(a/x) Proof by Mathematical Induction Proof by Transposition Proof by Construction Proof by Exhaustion Probabilistic Proof Combinatorial Proof Nonconstructive ProofEu...
Proof by mathematical induction gives rise to various kinds of eureka steps, e.g., missing lemmata and generalization. Most inductive theorem provers rely upon user intervention in supplying the required eureka steps. In contrast, we present a novel theorem-proving architecture for supporting the aut...
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