WRITING PROOFS Christopher Heil Georgia Institute of Technology
... true. But now, in addition, assume that statement B is false. Build on all this information until you obtain a contradiction. This means that your assumption that statement B is false is impossible. Here is an example. Theorem 2. If a positive integer m is evenly divisible by some integer n > 1, the ...
... true. But now, in addition, assume that statement B is false. Build on all this information until you obtain a contradiction. This means that your assumption that statement B is false is impossible. Here is an example. Theorem 2. If a positive integer m is evenly divisible by some integer n > 1, the ...
A(x)
... reasonable, for we couldn’t perform proofs if we did not know which formulas are axioms). It means that there is an algorithm that for any WFF given as its input answers in a finite number of steps an output Yes or NO on the question whether is an axiom or not. A finite set is trivially decidabl ...
... reasonable, for we couldn’t perform proofs if we did not know which formulas are axioms). It means that there is an algorithm that for any WFF given as its input answers in a finite number of steps an output Yes or NO on the question whether is an axiom or not. A finite set is trivially decidabl ...
lecture05
... [what 2nd order quantifiers did we need ?] – The graph is 3-colorable – The graph has a Hamiltonean path ...
... [what 2nd order quantifiers did we need ?] – The graph is 3-colorable – The graph has a Hamiltonean path ...
Series, Part 1 - UCSD Mathematics
... k=1 is bounded. Then, by Theorem 3.6(b), there exists a convergent subsequence. Since bnk + for all k 2 N, the subsequence converges to some number + . Since ...
... k=1 is bounded. Then, by Theorem 3.6(b), there exists a convergent subsequence. Since bnk + for all k 2 N, the subsequence converges to some number + . Since ...
Practical suggestions for mathematical writing
... as possible. It is OK to postpone definitions to a later “Notation” section (and to refer readers in the introduction to this later section) if those definitions are standard enough that most readers will be able to read the introduction without them. (14) Start induction arguments with base case n ...
... as possible. It is OK to postpone definitions to a later “Notation” section (and to refer readers in the introduction to this later section) if those definitions are standard enough that most readers will be able to read the introduction without them. (14) Start induction arguments with base case n ...
Computability - Homepages | The University of Aberdeen
... They all have cardinality 0 • One could easily think that all infinite sets have this cardinality: there would not exist a cardinality 1 • This turns out to be false. In fact, the real numbers are a larger set than N. – (In fact, for all sets A, the power set of A is larger than A) ...
... They all have cardinality 0 • One could easily think that all infinite sets have this cardinality: there would not exist a cardinality 1 • This turns out to be false. In fact, the real numbers are a larger set than N. – (In fact, for all sets A, the power set of A is larger than A) ...
Mathematics 220 Homework for Week 7 Due March 6 If
... Because m, n and m + 1 are positive, from the above inequality we conclude that m < n < m + 1. But there is no integer which is strictly between m and m + 1. This contradicts the assumption that n is an integer and proves the statement. 5.36 Let a, b ∈ R. Prove that if ab 6= 0, then a 6= 0 by using ...
... Because m, n and m + 1 are positive, from the above inequality we conclude that m < n < m + 1. But there is no integer which is strictly between m and m + 1. This contradicts the assumption that n is an integer and proves the statement. 5.36 Let a, b ∈ R. Prove that if ab 6= 0, then a 6= 0 by using ...
Hierarchical Introspective Logics
... announcement in June 1993 of a proof of the FLT conjecture ("Fermat's Last Theorem"). Once again a mathematical question that might conceivably have been unanswerable was being answered by the efforts of humans to find an acceptable proof of a standing conjecture. Earlier the famous 4 Color Conjectu ...
... announcement in June 1993 of a proof of the FLT conjecture ("Fermat's Last Theorem"). Once again a mathematical question that might conceivably have been unanswerable was being answered by the efforts of humans to find an acceptable proof of a standing conjecture. Earlier the famous 4 Color Conjectu ...
Fermat’s Last Theorem can Decode Nazi military Ciphers
... And even though the time period between these 2 events are 302 years apart, this type of logic parallels with the WWII Bletchley Park military headquarters in the UK when they were trying to crack the secret war codes using some form of deductive reasoning which stems from Euclid’s geometric laws. ...
... And even though the time period between these 2 events are 302 years apart, this type of logic parallels with the WWII Bletchley Park military headquarters in the UK when they were trying to crack the secret war codes using some form of deductive reasoning which stems from Euclid’s geometric laws. ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.