Lecture 9 21 127 Concepts of Math 09 17 2012 Administrivia Homework 3 is due on Thursday Don t use the online submission on CMG Print and hand to TA Don t cheat Sets Wrapup Russell s Paradox Bertrand Russell identified an issue with set theory in 1901 He defined the following set R x x x The set N is an example of an element of R that is N R There are some examples of non elements of R i e sets that contain themselves but they have to be weird the set of all non squirrels the set X 1 2 3 X the set of all sets With this definition we ask Is R R Or is R R Either way we have a contradiction Bertrand Russell discovered this paradox and sent it in a letter to G Frege just as Frege was completing Grundlagen der Arithmetik This invalidated much of the rigor of the work and Frege was forced to add a note at the end stating A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished I was put in this position by a letter from Mr Bertrand Russell when the work was nearly through the press What went wrong We don t want to allow this object to be a set We want our theory to not have such weird inconsistencies Some fixes were proposed later in 1908 by Russell and Ernst Zermelo This is the reason we should always specify a universal set when we use set builder notation Defining N We define 0 then 1 0 then 2 0 1 etc Definition Given any set X the successor of X denoted by S X is defined to be S X X X In general then this defines n 0 1 2 n 1 Definition A set I is called inductive provided 1 1 I 2 If n I then S n I as well Intuitively we know that N is inductive as is N 0 Z etc Definition The set of all natural numbers is the set of numbers that belong to every inductive set N I I S S is inductive Put another way N is the smallest inductive set 1 Note Why is the intersection above non empty That is how do we know there are actually any inductive sets to begin with Well we have to assume this We have to make an axiomatic assumption somewhere in our theory Stating the Principle Of Induction Theorem Principle of Mathematical Induction Let P n be some fact or observation that depends on the natural number n Assume that 1 P 1 is a true statement 2 For every k N if P k is true then we can conclude necessarily that P k 1 is true Then the statement P n must be true for every natural number n N Proof Define the set S to be the natural numbers for which the statement P is true That is define S n N P n is true By definition S N Furthermore the assumptions of the theorem guarantee that 1 S and that whenever k S we know k 1 S as well This means S is an inductive set and by the observation we made after defining N we know that N S Therefore S N so the statement P n must be true for every natural number n Logic Mathematical Statements Definition A mathematical statement or proposition or logical statement is a grammatically correct sentence or string of sentences composed of English words symbols and mathematical symbols that has exactly one truth value either True or False Example First some examples 142857 5 714285 1 2 0 For every n N X k k n n n 1 2 Any even natural number greater than or equal to 4 can be written as the sum of two prime numbers For any sets A and B if A B then P A P B For any two integers x y Z x y is odd if and only if x and y are both odd Second some non examples 1 2 5 Math is cool x 7 This sentence is false 2 Law Of The Excluded Middle The Law Of The Excluded Middle is the axiom of logic that states that a proposition can only have one truth value We have to agree on that Other areas of logic deal with different axioms Modal Logic for example allows statements to be kinda true in a sense It s possible to use the Excluded Middle to divide an argument into cases Here are two examples Claim Every integer x Z has remainder 0 or 1 when divided by 4 Proof Let x Z Then either x is odd or x is even If x is even then x 2k for some k Z so x2 4k 2 and leaves remainder 0 when divided by 4 If x is odd then x 2 1 for some Z so x2 2 1 2 4 2 4 1 4 2 1 and leaves remainder 1 when divided by 4 This covers all cases so the claim follows We were able to identify a property that is either true or false for all integers and considered each case separately In either case we reached the desired conclusion so it must be true The following is a more striking example of this law The strange thing about the Law of the Excluded Middle is it doesn t tell us how to determine whether a statement is True or False we are just guaranteed that it s one way or the other Sometimes that s the best we can do Claim There are real numbers a and b that are both irrational such that ab is rational 2 How can you prove that The number 2 is either Proof We know 2 is irrational Question Why 2 and b 2 is the example we seek Otherwise it is rational or irrational If it s rational then a 2 irrational and we can use a 2 and b 2 as the example we seek because 2 2 2 ab 2 2 2 2 2 2 This is an example of a non constructive proof It tells us something exists and narrows it down to two possibilities even without actually telling us exactly what it is It is a direct use of the Law of the Excluded 2 Middle that causes this Question Can you prove somehow that 2 is irrational Apparently it is true but there is no known simple proof of this fact Maybe you can find one Defining Propositions We will frequently assign letter names to propositions so we can refer to them easily To define these we have to be careful with how we use quotation marks They should encompass the statement and only the statement Some good examples Let P be 2 is irrational Let Q be the statement There are infinitely many prime numbers Some bad examples Let P be 2 is irrational Let Q There are infinitely many prime …