PROOFS BY INDUCTION AND CONTRADICTION, AND WELL
... 1.1. Strong induction. We can also use the principle of induction to prove a similar result, variously called the ‘principle of complete induction’ or the ‘principle of strong induction.’ Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 0 ∈ S, an ...
... 1.1. Strong induction. We can also use the principle of induction to prove a similar result, variously called the ‘principle of complete induction’ or the ‘principle of strong induction.’ Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 0 ∈ S, an ...
Lecture I: Dirac Notation
... • Put a bra and a ket together to get a cnumber – %"|#! := a c-number ...
... • Put a bra and a ket together to get a cnumber – %"|#! := a c-number ...
PROOFS BY INDUCTION AND CONTRADICTION, AND WELL
... 1.1. Strong induction. We can use the principle of induction to prove a similar result, variously called the ‘principle of complete induction’ or the ‘principle of strong induction.’ Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 1 ∈ S, and (ii ...
... 1.1. Strong induction. We can use the principle of induction to prove a similar result, variously called the ‘principle of complete induction’ or the ‘principle of strong induction.’ Proposition (Principle of strong induction). If S ⊂ N is a subset of the natural numbers such that (i) 1 ∈ S, and (ii ...
ANALYSIS I A Number Called e
... These supplementary notes by H A Priestley provide a (non-examinable) proof of the useful fact that ...
... These supplementary notes by H A Priestley provide a (non-examinable) proof of the useful fact that ...
proof - Jim Hogan
... Inspiration from Dr. Paul Brown, Carmel School, WA Paul presented a similar workshop at MAV Conference, 2008. I enjoyed the session so much I decided then to present my version here in Palmerston North. Thanks also to a little book called Q.E.D. Beauty in Mathematical Proof written by Burkard Polste ...
... Inspiration from Dr. Paul Brown, Carmel School, WA Paul presented a similar workshop at MAV Conference, 2008. I enjoyed the session so much I decided then to present my version here in Palmerston North. Thanks also to a little book called Q.E.D. Beauty in Mathematical Proof written by Burkard Polste ...
Mathematical Induction - Singapore Mathematical Society
... called Pascal's Triangle. The first is a Chinese version copied from a diagram that appeared in the Ssu-yii.an yii.-chien (Precious Mirror of the Four Elements) by Chu Shih-chieh in 1303. Chu disclaims credit for the triangle and it seems likely that it originated in China about 1100. Note the use o ...
... called Pascal's Triangle. The first is a Chinese version copied from a diagram that appeared in the Ssu-yii.an yii.-chien (Precious Mirror of the Four Elements) by Chu Shih-chieh in 1303. Chu disclaims credit for the triangle and it seems likely that it originated in China about 1100. Note the use o ...
MATH 327 - Winona State University
... 1. Binary Operations on Sets 2. Basic Introduction to Groups 3. Basic Introduction to Rings Remarks: The emphasis, during all topics of the course, is learning how to write proofs. This involves significant feedback from the instructor, but also possibly from peers. In addition to creating proof ...
... 1. Binary Operations on Sets 2. Basic Introduction to Groups 3. Basic Introduction to Rings Remarks: The emphasis, during all topics of the course, is learning how to write proofs. This involves significant feedback from the instructor, but also possibly from peers. In addition to creating proof ...
Mathematical Logic
... confirm them with experimental evidence. Sampling: we analyse many pieces of evidence statistically and identify patterns. Mathematics: we derive conclusions based on mathematical proof. ...
... confirm them with experimental evidence. Sampling: we analyse many pieces of evidence statistically and identify patterns. Mathematics: we derive conclusions based on mathematical proof. ...
Solutions 1
... where the “bar” means any element from Ak which differs from akk (∀k ∈ N). From this argumentation, one can see that no assumption on the cardinality (finite vs infinite) of the sets A1 , A2 , A3 , ... was made. And in fact, they need not be infinite. As long as each of the sets has at least two ele ...
... where the “bar” means any element from Ak which differs from akk (∀k ∈ N). From this argumentation, one can see that no assumption on the cardinality (finite vs infinite) of the sets A1 , A2 , A3 , ... was made. And in fact, they need not be infinite. As long as each of the sets has at least two ele ...
A Brief Note on Proofs in Pure Mathematics
... You should note that induction can only be applied when the statements are indexed by the natural numbers. There has to be a first case (the base case), and you have to have be able to move to the ‘next’ statement (the induction step). If the statements are indexed by real numbers, then there is no ...
... You should note that induction can only be applied when the statements are indexed by the natural numbers. There has to be a first case (the base case), and you have to have be able to move to the ‘next’ statement (the induction step). If the statements are indexed by real numbers, then there is no ...
Primitive Recursive Arithmetic and its Role in the Foundations of
... Mathematica and in reaction to that work. His specific complaint about the foundations of arithmetic (i.e. number theory) in that work was, as implied by his title, the essential role in it of logic and in particular quantification over infinite domains, even for the understanding of the most elemen ...
... Mathematica and in reaction to that work. His specific complaint about the foundations of arithmetic (i.e. number theory) in that work was, as implied by his title, the essential role in it of logic and in particular quantification over infinite domains, even for the understanding of the most elemen ...
report
... which remains an open question is which γ have γr equidistibuted. We do know, however, that it is at least not true for all irrational γ. Consider Binet’s formula for the nth Fibonacci number √ !n √ !n ...
... which remains an open question is which γ have γr equidistibuted. We do know, however, that it is at least not true for all irrational γ. Consider Binet’s formula for the nth Fibonacci number √ !n √ !n ...
Lecture 3
... 2 are all defined. Defining R takes some work. It turns out that R is ‘essentially’ the unique set of numbers which satisfies (P1) - (P12) together with an additional axiom (P13) which ensures there are ‘no holes’ or ‘gaps’ in our number line. This will be discussed in later lectures. Definition (3. ...
... 2 are all defined. Defining R takes some work. It turns out that R is ‘essentially’ the unique set of numbers which satisfies (P1) - (P12) together with an additional axiom (P13) which ensures there are ‘no holes’ or ‘gaps’ in our number line. This will be discussed in later lectures. Definition (3. ...
CPSC 311: Analysis of Algorithms Proof by Induction Example
... CPSC 311: Analysis of Algorithms Proof by Induction Example Problems These problems are solely for your own bene t in practicing inductive proofs. They will not be turned in or graded. A solution sheet will not be handed out. If you want to discuss whether you are doing them correctly, you can meet ...
... CPSC 311: Analysis of Algorithms Proof by Induction Example Problems These problems are solely for your own bene t in practicing inductive proofs. They will not be turned in or graded. A solution sheet will not be handed out. If you want to discuss whether you are doing them correctly, you can meet ...
infinity - Department of Mathematics | University of Washington
... than rational numbers, between any two real numbers you can always find a rational one • There are surfaces with finite volume but infinite surface area (but not the opposite!) ...
... than rational numbers, between any two real numbers you can always find a rational one • There are surfaces with finite volume but infinite surface area (but not the opposite!) ...
Mathematical Logic Deciding logical consequence Complexity of
... The modern notion of symbolic formal proof was developed in the 20th century by logicians and mathematicians such as Russell, Frege and Hilbert. The benefit of formal logic is that it is based on a pure syntax: a precisely defined symbolic language with procedures for transforming symbolic statement ...
... The modern notion of symbolic formal proof was developed in the 20th century by logicians and mathematicians such as Russell, Frege and Hilbert. The benefit of formal logic is that it is based on a pure syntax: a precisely defined symbolic language with procedures for transforming symbolic statement ...
A simple proof of Parsons` theorem
... ago, Charles Parsons showed that the provably total functions of IΣ1 are exactly the primitive recursive functions. In this paper, we observe that Parsons’ result is a consequence of Herbrand’s theorem concerning the ∃∀∃-consequences of universal theories. We give a self-contained proof requiring on ...
... ago, Charles Parsons showed that the provably total functions of IΣ1 are exactly the primitive recursive functions. In this paper, we observe that Parsons’ result is a consequence of Herbrand’s theorem concerning the ∃∀∃-consequences of universal theories. We give a self-contained proof requiring on ...
Assignment 2: Proofs
... (a) Give a direct proof that: If x is an integer then x3 − x is divisible by 3. (b) Give a contrapositive proof that: Let x and y be real numbers. If the product x · y is irrational then either x or y is an irrational number. ...
... (a) Give a direct proof that: If x is an integer then x3 − x is divisible by 3. (b) Give a contrapositive proof that: Let x and y be real numbers. If the product x · y is irrational then either x or y is an irrational number. ...
1 Introduction 2 What is Discrete Mathematics?
... relations, counting, probability and graph theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. Some of these applications are listed below. • The analysis of algorithms requires the ability to count the number of operations in an alg ...
... relations, counting, probability and graph theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. Some of these applications are listed below. • The analysis of algorithms requires the ability to count the number of operations in an alg ...