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MATH135 Lecture Notes.pdf

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MATH 135
Janelle Resch

MATH135 Lecture Notes (1-4) Erin Edward Introduction Question: What is mathematics?  Math seeks out patterns to try and seek out reality based on true statements. Statements are either true or false.  Math focuses on proving the truth (or falsity) of a statement through formal logical steps. Question: How does one describe reality through logical statements? What is a proof?  A proof is a mathematical statement that has no ambiguity (no uncertainty) and is either true or false. In math, we make precise use of the language by stating and proving results. Question: How does one write a proof as a true, precise statement? What tools does one need to prove statements properly?  Language  Notation, presentation, and format  Arguments (knowledge) are not definitions  Precision and logic Statements REMARK: Good math uses complete sentences and proper punctuation. Symbols are only used for convenience (to make math much easier to read and to comprehend). Don’t overuse symbols (know precisely why they are being used). A conditional statement or an open sentence is some sort of phrase based on something else. An open statement is a statement using variables that is either true or false once values from the same domain (a set of allowable or legal values) are given for the variables. Ex: you are taller than me Not enough information Ex: z =49 What is z? Ex: x>0 but what is x? This statement relies on something else. It needs “x ∈ Z” Ex: if a, b ∈ Z and a divides b and b divides c… Not precise enough because c is not defined. Everything must be introduced. If a, b, c ∈ Z and a divides b and b divides c, then a divides c NOTE: ∈ means “is an element of” Z means the integers MATH135 Lecture Notes (1-4) Erin Edward REMARK:  The first step of proving or disproving a statement is to decide if the statement is true or false.  If false, use an example to disprove.  Could you convince your grandma? Ex: An example of a true statement is 2+2=4 OR x -1=0 Two distinct real roots: -1 and 1 Ex: if x and y are even integers, (x+y) is an odd integer This is a false statement Ex: 2+2=5 is a false statement OR x +1=0 (2 imaginary roots) What are the outcomes of the following statements? a) A Λ B TRUE A: Paris is the capital of France (T) B: 2+2=4 (T) b) A Λ B FALSE A : 6 is prime (F) B : 2 is odd (F) c) A: Paris is the capital of France ¬A: A Paris is NOT the capital of France NOTE: ¬(Λ)=(V) ¬(V)=(Λ) Ex: A: I am a male (F) B: 2 is prime (T) A V B is TRUE The most common statement we will be considering this class is implication, which is the statement of the form: If A (hypothesis), then B (conclusion) OR A→B Where A and B are statements REMARK: an implication is a compound statement, that is to say a statement composed of at least 1 statement and 1 connective (¬, Λ, V, →, ↔) MATH135 Lecture Notes (1-4) Erin Edward 𝑥 −1 EX: If x∈R, solve = 10 𝑥+1 By factoring, we obtain → x-1=10 →x=11 x∈ I NOTE: If expression does not simplify so nicely, must state restrictions. → means “implies” + Ex: IF r, n ∈ Z , 1 ≤ 𝑟 ≤ 𝑛, then NOTE: this is the binomial sum, part of the binomial theorem where Binomial coefficient LS: = (𝑛+1 ! 𝑟! 𝑛−𝑟+1 ! 𝑛! 𝑛! RS: = + (𝑟−1 ! 𝑛−𝑟∓1 ! 𝑟! 𝑛−𝑟 ! = (𝑟 𝑛! + (𝑛−𝑟+1 𝑛! (𝑟 𝑟−1 ! 𝑛−𝑟+1 !) 𝑛−𝑟+1 𝑟 𝑟−1 ! 𝑛−𝑟 !) 𝑟𝑛! 𝑛! 𝑛−𝑟+1 ) = ( ) + ( ) 𝑟! 𝑛−𝑟+1 ! 𝑟! 𝑛−𝑟+1 ! =𝑟𝑛!+𝑛!(𝑛−𝑟+1) 𝑟! 𝑛−𝑟+1 ! 𝑛!(𝑟+𝑛−𝑟+1) = 𝑟! 𝑛−𝑟+1 ! 𝑛! 𝑛+1 ) = 𝑟! 𝑛−𝑟+1 ! (𝑛+1 ! =𝑟! 𝑛−𝑟+1 ! Thus LS=RS NOTE: 0!=1 by definition Ex: Let x∈R 2 if x<0, then x +1>0 Lets assume x<0 ∀ x∈R Notice that regardless of x<0 or x>0, x >02 2 So, x +1>0 Since x<0, x≠0 NOTE: if __________, then __________ is always implication ∀ means “for all of” MATH135 Lecture Notes (1-4) Erin Edward Write the hypothesis and conclusion of the following compound statements: a) If x is a positive real number (hypothesis), then the logx>0 (conclusion) b) Let f(x)=sinx (hypothesis), then f(x)=x for some real number x with 0≤ 𝑥 ≤2𝜋 (conclusion) DEFINTION: An even integer is m, m has the form m=2p where p∈Z An odd integer is n, n has the form n=2q-1 where q∈Z DEFINITION: An integer n divides an integer m (denoted by n|m) if m=nk, for some k∈Z DEFINITION: An integer p is prime if the only positive integers that divide p is 1 and p DIRECT PROOFS: Given two statements A and B, if we assume that A is true and we want to prove A→B, we must show that B is true using the objects from the hypothesis. Ex: If x∈Z and x is even, then x2 is even. Assume our hypothesis is true. If x is an even integer, by definition x=2p, p∈Z 2 ASSIDE: Want to prove x =2q, q∈Z 2 2 Then x =(2p) =4p 2 2 =2(2p ) Since p∈Z, then 2p ∈Z 2 Thus, by definition, x is even. Ex: Let A and B be sets. Then (A-b)=A∩B c c c c If we want to prove that (A-B)=A∩B , we must show that (A-B)c(A∩B ) and (A∩B )c(A-B) NOTE: This is an “if and only if” proof First lets prove (A-B)c(A∩B ) c c Let x∈(A-B), then x∈A and xɆB by definition (want to define A∩B ) We also want to prove that (A∩B )c(A-B) (want to define A-B) c Let x∈ (A∩B ) Then x∈A and xɆB → x∈(A-B) by definition We want to prove that (A-B)c(A∩B ) and (A∩B )c(A-B) c ↔Let x∈(A-B) ↔Then x∈A and x∈B by definition c ↔ x∈ (A∩B ) Ex: Let x∈, if x<0 then x +1>0 2 Notice x +1>0 If x=0 → x +1=0+1>0 If x<0 (a∈R) → x +1=(-a) +1>0 2 2 If x>0 (b∈R) → x +1=(b) +1>0 MATH135 Lecture Notes (1-4) Erin Edward Ex: Solve |2x-5|=3 NOTE: Absolute value is the distance from 0 on a number line By definition we have that 1. 2x-5=3 → 2x=8 2. 2x-5=-3 → 2x=2 Then #1 becomes x=4 and #2 becomes x=1 Ex: Solve |x-5|<2 This is an equivalent to saying →-2 or ≤ instead of < An example of a statement with ‘nested quantifiers’ is… DEFINITION: The LIMIT of f(x) as x -> a equals L means that ∀ε>0, ∃δ>0 NOTE: ε is epsilon δ is delta 0 < |x-a|a(x)=L) |x-a|amx+b)=ma+b Let ε>0 be a real number We want to show ∀ε>0 ∃ δ >0 -> 0a+b=ma+b by definition MATH135 Lecture Notes (1-4) Erin Edward Define: the limit of f(x) as x->a=L x->a(x)=L) means that ∀ε0 ∃ δ>0 0a Let ε>0 be a real number We want to show ∀𝜀>0 ∃ δ>0 : 0 f(x)=0 Let ε>0 be a real number We want to show that ∀ε>0 ∃ δ>0 : 0
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