Quantifiers, Proofs - Department of Mathematics
... On the other hand, if ∀x P (x) is false then it is not true that for every x, P (x) holds, hence for some x, P (x) must be false. Thus: ∀x P (x) ≡ ∃x P (x) . This two rules can be applied in successive steps to find the negation of a more complex quantified statement, for instance: ∃x∀y p(x, y) ≡ ∀x ...
... On the other hand, if ∀x P (x) is false then it is not true that for every x, P (x) holds, hence for some x, P (x) must be false. Thus: ∀x P (x) ≡ ∃x P (x) . This two rules can be applied in successive steps to find the negation of a more complex quantified statement, for instance: ∃x∀y p(x, y) ≡ ∀x ...
Lecture22 – Finish Knaves and Fib
... What are they? 1. Assume what you are proving, 2. plug in definitions, 3. do some work, 4. show the opposite of what you are proving (a contradiction). 1. Assume the opposite of what you are ...
... What are they? 1. Assume what you are proving, 2. plug in definitions, 3. do some work, 4. show the opposite of what you are proving (a contradiction). 1. Assume the opposite of what you are ...
Version of Gödel`s First Incompleteness Theorem
... Self-reference can be made simpler still • modern proof of non-existence of halting testers is very simple ⇒ negation of halting is not recursively enumerable • provability is recursively enumerable by definition ⇒ incompleteness is obtained, if halting is automatically represented as a formula • th ...
... Self-reference can be made simpler still • modern proof of non-existence of halting testers is very simple ⇒ negation of halting is not recursively enumerable • provability is recursively enumerable by definition ⇒ incompleteness is obtained, if halting is automatically represented as a formula • th ...
slides04-p - Duke University
... • Case 2: x has a prime factor p. But if pn, then p mod x = 1. So p>n, and we’re done. CompSci 102 ...
... • Case 2: x has a prime factor p. But if pn, then p mod x = 1. So p>n, and we’re done. CompSci 102 ...
Structural Induction - Department of Computer Science
... However, we know from CSC165 this is more complicated than necessary. Therefore, instead of counting each elementary step it is sufficient to count each “chunk” of instructions as 1 step, where “chunk” is a sequence of instructions that always gets executed together in constant time. Algorithm compl ...
... However, we know from CSC165 this is more complicated than necessary. Therefore, instead of counting each elementary step it is sufficient to count each “chunk” of instructions as 1 step, where “chunk” is a sequence of instructions that always gets executed together in constant time. Algorithm compl ...
The strong law of large numbers - University of California, Berkeley
... others and deals with the case of identically distributed, independent random variables (r.v.'s). In this case it is known, after Kolmogorov,1 that a nasc for the validity of the SLLN is the finiteness of the first absolute moment of the common distribution function (d.f.). For use in certain statis ...
... others and deals with the case of identically distributed, independent random variables (r.v.'s). In this case it is known, after Kolmogorov,1 that a nasc for the validity of the SLLN is the finiteness of the first absolute moment of the common distribution function (d.f.). For use in certain statis ...
Supplement: The Fundamental Theorem of Algebra - Faculty
... method was not to calculate a root but to demonstrate its existence. . . . Gauss gave three more proofs of the theorem.” ...
... method was not to calculate a root but to demonstrate its existence. . . . Gauss gave three more proofs of the theorem.” ...
Series, Part 1 - UCSD Mathematics
... Recall from Theorem 3.11 that a sequence of real numbers converges if and only if it is a Cauchy sequence. Given a sequence fan g, we apply P this theorem to the corresponding sequence of partial sums fsn g to get the following. The series 1 n=1 an converges if and only if fsn g is a Cauchy sequence ...
... Recall from Theorem 3.11 that a sequence of real numbers converges if and only if it is a Cauchy sequence. Given a sequence fan g, we apply P this theorem to the corresponding sequence of partial sums fsn g to get the following. The series 1 n=1 an converges if and only if fsn g is a Cauchy sequence ...
Mathematical Reasoning (Part III)
... An element belonging to some prescribed set D and possessing a certain property P is unique if it is the only element of D having property P . Typical ways to prove uniqueness: 1. By a direct proof: Assume that x and y are elements of D possessing property P and show that x = y. 2. By a proof by con ...
... An element belonging to some prescribed set D and possessing a certain property P is unique if it is the only element of D having property P . Typical ways to prove uniqueness: 1. By a direct proof: Assume that x and y are elements of D possessing property P and show that x = y. 2. By a proof by con ...
On two problems with the Theory of the Creating Subject
... our minds are taken to be essential to what a mathematical construction is, and others are not. The stance that Brouwer assumes is of a kind with Turing’s, who devised his theoretical analysis of (mechanical) computation in terms of an idealised human, not a machine; Gandy proposed the term ‘comput ...
... our minds are taken to be essential to what a mathematical construction is, and others are not. The stance that Brouwer assumes is of a kind with Turing’s, who devised his theoretical analysis of (mechanical) computation in terms of an idealised human, not a machine; Gandy proposed the term ‘comput ...
An Example of Induction: Fibonacci Numbers
... fifth Fibonacci number is a multiple of 5. As usual in mathematics, we have to start by carefully defining the objects we are studying. Definition. The sequence of Fibonacci numbers, F0 , F1 , F2 , . . ., are defined by the following equations: F0 = 0 F1 = 1 Fn + Fn+1 = Fn+2 Theorem 1. The Fibonacci ...
... fifth Fibonacci number is a multiple of 5. As usual in mathematics, we have to start by carefully defining the objects we are studying. Definition. The sequence of Fibonacci numbers, F0 , F1 , F2 , . . ., are defined by the following equations: F0 = 0 F1 = 1 Fn + Fn+1 = Fn+2 Theorem 1. The Fibonacci ...
CS311H: Discrete Mathematics Mathematical Proof Techniques
... Here, proof by cases is useful because definition of absolute value depends on whether number is negative or not. ...
... Here, proof by cases is useful because definition of absolute value depends on whether number is negative or not. ...
methods of proofs
... What if I prove “If √r is rational, then r is rational”, is it equivalent? Yes, this is equivalent, because it is the contrapositive of the statement, so proving “if P, then Q” is equivalent to proving “if not Q, then not P”. ...
... What if I prove “If √r is rational, then r is rational”, is it equivalent? Yes, this is equivalent, because it is the contrapositive of the statement, so proving “if P, then Q” is equivalent to proving “if not Q, then not P”. ...
Slides for Rosen, 5th edition
... • Now, just from this, what do we know about x and y? You should think back to the definition of rational: • … Since x is rational, we know (from the very definition of rational) that there must be some integers i and j such that x = i/j. So, let ix,jx be such integers … • We give them unique names ...
... • Now, just from this, what do we know about x and y? You should think back to the definition of rational: • … Since x is rational, we know (from the very definition of rational) that there must be some integers i and j such that x = i/j. So, let ix,jx be such integers … • We give them unique names ...
Notes - Cornell Computer Science
... to a computational treatment that is simultaneously very useful and practical and yet completely precise mathematically. It is this dual character that we are exploring in this course. Last year we used these reals to do some computational geometry. They are becoming increasingly important in the st ...
... to a computational treatment that is simultaneously very useful and practical and yet completely precise mathematically. It is this dual character that we are exploring in this course. Last year we used these reals to do some computational geometry. They are becoming increasingly important in the st ...
The Foundations: Logic and Proofs
... repeatedly apply the transformation T, we will eventually reach the integer 1. For example, starting with x = 13: T(13) = 3∙13 + 1 = 40, T(40) = 40/2 = 20, T(20) = 20/2 = 10, T(10) = 10/2 = 5, T(5) = 3∙5 + 1 = 16,T(16) = 16/2 = 8, T(8) = 8/2 = 4, T(4) = 4/2 = 2, T(2) = 2/2 = 1 ...
... repeatedly apply the transformation T, we will eventually reach the integer 1. For example, starting with x = 13: T(13) = 3∙13 + 1 = 40, T(40) = 40/2 = 20, T(20) = 20/2 = 10, T(10) = 10/2 = 5, T(5) = 3∙5 + 1 = 16,T(16) = 16/2 = 8, T(8) = 8/2 = 4, T(4) = 4/2 = 2, T(2) = 2/2 = 1 ...
Squares in arithmetic progressions and infinitely many primes
... showed me a completely different proof of Theorem 2, this time relying on one of the most influential results in algebraic and arithmetic geometry, Faltings’ theorem [3]. Faltings’ theorem is not easy to state, requiring a general understanding of an algebraic curve and its genus. The basic idea is ...
... showed me a completely different proof of Theorem 2, this time relying on one of the most influential results in algebraic and arithmetic geometry, Faltings’ theorem [3]. Faltings’ theorem is not easy to state, requiring a general understanding of an algebraic curve and its genus. The basic idea is ...