Counting Subsets of a Set
... Theorem: Suppose a set contains n1 indistinguishable elements of one type, n2 indistinguishable elements of another type, and so on, through k types, where n1 + n2 + … + nk = n. Then the number of (distinguishable) permutations of the n elements is n!/(n1!n2!…nk!). ...
... Theorem: Suppose a set contains n1 indistinguishable elements of one type, n2 indistinguishable elements of another type, and so on, through k types, where n1 + n2 + … + nk = n. Then the number of (distinguishable) permutations of the n elements is n!/(n1!n2!…nk!). ...
Proof by Induction
... some other number(s) smaller than n. This is the hard part. 3. Look for holes. Look for cases where your inductive argument breaks down. Solve those cases directly. Don’t be clever here; be stupid but thorough. 4. Rewrite everything. Your first proof is a rough draft. Rewrite the proof so that your ...
... some other number(s) smaller than n. This is the hard part. 3. Look for holes. Look for cases where your inductive argument breaks down. Solve those cases directly. Don’t be clever here; be stupid but thorough. 4. Rewrite everything. Your first proof is a rough draft. Rewrite the proof so that your ...
1. Proof Techniques
... combine this with other known facts F and theorems T to conclude that Q must be true. Think of P as the starting point, and Q the destination. The other known facts F and theorems T make up the route that go from P to Q. Each step of the route must be logically connected. In your draft, you can work ...
... combine this with other known facts F and theorems T to conclude that Q must be true. Think of P as the starting point, and Q the destination. The other known facts F and theorems T make up the route that go from P to Q. Each step of the route must be logically connected. In your draft, you can work ...
Beginning Deductive Logic
... This is rather a deep and tricky question, as of course are relatives like: “What is physics?” and “What is economics?” and “What is art?”. Perhaps one has to be either brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to cr ...
... This is rather a deep and tricky question, as of course are relatives like: “What is physics?” and “What is economics?” and “What is art?”. Perhaps one has to be either brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to cr ...
![arXiv:math/0407326v1 [math.CO] 19 Jul 2004](http://s1.studyres.com/store/data/016678102_1-7a26a9b4445f7030b68419a5be89f8d8-300x300.png)
4 slides/page
... It’s not that bad. • Because of the way we defined ⇒, A1 ∧ . . . ∧ An ⇒ B is guaranteed to be true if A1 ∧ . . . ∧ An is false. • But if A1 ∧ . . . ∧ An is true, B is true, since the argument is valid. • Thus, (A1 ∧ . . . ∧ An) ⇒ B is a tautology. For the converse, suppose (A1 ∧ . . . ∧ An) ⇒ B is a ...
... It’s not that bad. • Because of the way we defined ⇒, A1 ∧ . . . ∧ An ⇒ B is guaranteed to be true if A1 ∧ . . . ∧ An is false. • But if A1 ∧ . . . ∧ An is true, B is true, since the argument is valid. • Thus, (A1 ∧ . . . ∧ An) ⇒ B is a tautology. For the converse, suppose (A1 ∧ . . . ∧ An) ⇒ B is a ...
![[Part 1]](http://s1.studyres.com/store/data/008795785_1-ac97e8bcbccd58a3457a6ac11081a546-300x300.png)
[Part 1]
... Other possibilities suggest themselves. Letting n - 3k | (n - k + 1, 2k + 1) gives again n - 3 k | n - k + l whence also n - 3k j 3(n - k + 1) - (n - 3k) o r n - 3k | 2n + 3. If we then take 2n + 3 = prime we again obtain a useful theorem. It seems clear from just these samples that the theorems of ...
... Other possibilities suggest themselves. Letting n - 3k | (n - k + 1, 2k + 1) gives again n - 3 k | n - k + l whence also n - 3k j 3(n - k + 1) - (n - 3k) o r n - 3k | 2n + 3. If we then take 2n + 3 = prime we again obtain a useful theorem. It seems clear from just these samples that the theorems of ...
Dialetheic truth theory: inconsistency, non-triviality, soundness, incompleteness
... extend mathematical investigation into inconsistent domains.11 L et us therefore proceed with the assumption, and note that, because PA* has a recursive proof relation (PA*derivations form a recursive set of sequences of strings on the alphabet of L), and because all recursive relations can be repre ...
... extend mathematical investigation into inconsistent domains.11 L et us therefore proceed with the assumption, and note that, because PA* has a recursive proof relation (PA*derivations form a recursive set of sequences of strings on the alphabet of L), and because all recursive relations can be repre ...
PPT
... A proof of Q from H1, H2, … Hk is finite sequence of propositional forms Q 1, Q 2, … Qn such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by li ...
... A proof of Q from H1, H2, … Hk is finite sequence of propositional forms Q 1, Q 2, … Qn such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by li ...
PDF
... 3. A set S of formulas is called decidable if the set of Gödel numbers of S is decidable, i.e. if the characteristic function of that set is computable. 4. T is called axiomatizable, if there is a decidable subset of T whose logical consequences are exactly the theorems of T . T is finitely axiomat ...
... 3. A set S of formulas is called decidable if the set of Gödel numbers of S is decidable, i.e. if the characteristic function of that set is computable. 4. T is called axiomatizable, if there is a decidable subset of T whose logical consequences are exactly the theorems of T . T is finitely axiomat ...
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.