NOTES WEEK 01 DAY 1 SCOT ADAMS Question: What is Mathematics? Answer: The study of absolute truth. Question: Why is it so hard to teach and to learn? Answer: One must learn to play a variety of games called theorems. One must also learn a language to describe strategies for winning those games. The strategies have to be perfect, and the writing has to have no ambiguities whatsoever. Propositional Logic: Truth tables. We proved, via a truth table, for any propositions P and Q, that: r P ñ Q s ô r Q or pnot P q s. Unassigned HW: Show that pp &Qq ô pq&p q. Unassigned HW: Show that rpp ñ Qq&pQ ñ Rqs ñ rp ñ Rs. Convention: We will use Propositional Logic results in our proofs, typically without comment. Abbreviations: @ = for all, D = there exists, ñ = implies, ô = iff = if and only if, s.t. = such that. NOTE TO SELF: Introduce an object denoted which is NOT an element of any set. Any @ will refer to all objects that are NOT equal to. So, for example, the statement, @x, x is true. Similarly, D also refers only to objects that are NOT equal to. So, for example, Dx s.t. x 2 4 is understood to mean that there exists an object x such that both x and x 2 4 are true. Note: (if P, then Q) = (P ñ Q) Fact: @x, ( x x ). Fact: ( If two things are equal to the same thing, then they must be equal to each other. ) @x, y, z, ( rpx zq&py zqs ñ rx ys ). Date: September 6, 2016 Printout date: November 28, 2016. 1
2 SCOT ADAMS A null true statement: For any x, for any proposition P, we have: px xq ñ P. Basic Set Thory: Fact: 1 P t1, 2, 3u and 4 R t1, 2, 3u. Definition: Let P and Q be sets. Then P Ď Q means: @x, px P P q ñ px P Qq. Note: @x, px P P q ñ px P Qq can be simplified to @x P P, x P Q. Fact: @P, Q, if P and Q are sets, then pp Qq ô ppp Ď Qq&pQ Ď P qq. Note: @P, Q, if P and Q are sets, then pp Qq ô ppp Ď Qq&pQ Ď P qq can be simplified to @ sets P, Q, r pp Qq ô ppp Ď Qq&pQ Ď P qq s. Definition: H : tx x xu t u. Definition: Let P and Q be sets. Then P Y Q : tx px P P q or px P Qqu, P X Q : tx px P P q and px P Qqu, and P zq : tx px P P q and pnot px P Qqqu. Definitions: N : t1, 2, 3,...u tpositive integersu, N 0 : t0, 1, 2, 3,...u tsemipositive integersu, Z : t..., 3, 2, 1, 0, 1, 2, 3,...u tintegersu, Q : tm{n m P Z, n P Nu trational numbersu, R : treal numbersu, R : R Y t 8, 8u textended real numbersu, C : tcomplex numbersu, Note: Any coordinatized line is in 1-1 correspondene with R.
NOTES WEEK 01 DAY 1 3 Any coordinatized plane is in 1-1 correspondence with C. Note: In this class, number will typically mean real number. Note: N Ď N 0 Ď Z Ď Q Ď R Ď R. Also, R Ď C. Convention: We understand the binary operators `,,, { on R. Avoid ˆ and. Convention: We understand the binary Boolean operators ă, ą, ď, ě on R. Convention: The basic properties of `,,, {, ă, ą, ď, ě will be used, as needed, without proof. For example, there s no need to prove the statement: @a, b, c P R, pa ď bq ñ pa ` c ď b ` cq. Bounded intervals: pa, bq, ra, bq, pa, bs, ra, bs. Unbounded intervals: p 8, aq, p 8, as, pa, 8q, ra, 8q. Remark: The interval r1, 1s is equal to t1u. This kind of interval is called a degenerate interval. A degenerate interval is an interval containing only one number. Remark: Every interval is nonempty. Every interval is a subset of R. We do define, for example, r3, 8s : tx P R 3 ď x ď 8u, but call this an extended interval; it is not an interval. Every extended interval is a subset of R. The handout is our main reference for rules in writing proofs. Note: A key focus of the handout is on binding variables. Note: The past tense of the verb to bind is bound. The past tense of the verb to bound is bounded. We bind variables, after which they become bound, NOT bounded. (The misuse of bounded as a past tense of to bind is a pet peeve of mine!) We sometimes bound variables and functions, after which they become bounded. However, there is no discussion of bounding variables and/or functions in the handout. In the handout, we only use to bind and bound, never to bound and bounded. Note: An unbound variable is sometimes called a free variable.
4 SCOT ADAMS Verboten: a ` b b ` a This is incorrect, because the variables a and b are free. Do not use any variable until it is bound. There are only a few ways to bind variables and we will be discussing all of them in depth. Using a variable before it is bound is strictly forbidden and can result in loss of credit. The use of even just one free variable in a proof can, in some cases, render the entire proof invalid, and can result in a zero score. Correct: @a, b P R, a ` b b ` a. Note: There are exactly two quantifiers. They are: @ and D. The universal quantifier is @. The existential quantifier is D. Note: The use of a quantifier before a variable causes the variable to become bound until the end of the sentence. Consider the sentence: @a, b P R, a ` b b ` a. In this sentence, the variables a and b are bound from the @ to the period. It s not acceptable to write: @a, b P R, a`b b`a. Also, ab ba. Here, the a and b in ab ba are free variables, which is not allowed. On the other hand, it s fine to write: @a, b P R, a ` b b ` a. Also, @a, b P R, ab ba. Other methods of binding: In ta P R a 2 ď 1u r 1, 1s, the variable a is bound from t to u. In lim añ 2 a2 4, the variable a is bound from lim to. In max apr 2,1s In max app 2,1s In sup app 2,1s a 2 4, the variable a is bound from max to. a 2, the variable a is bound from max to. a 2 4, the variable a is bound from sup to. Similar remarks for min and inf. Note that supremum sup lub least upper bound. Note that infimum inf glb greatest lower bound. Still more methods of binding: The sentence Let x : 1 binds x until the end of the current section of proof. The sentence Given x P S binds x until the end of the current section of proof. The sentence Choose x P S binds x until the end of the current section of proof.
NOTES WEEK 01 DAY 1 5 Next goal: Prove that @ε ą 0, Dδ ą 0 s.t., @x P R, p0 ă x ă 2δq ñ px ` x 2 ă εq. This is a triply quantified statement because of the quantifiers @, then D, then @. By interpreting this statement as a game, we understood exactly why the statement is true. Game rules: You choose ε ą 0. I choose δ ą 0. You choose x P R satisfying 0 ă x ă 2δ. We check to see if px ` x 2 ă εq is true. If it is, I win. If not, you win. We need to develop a strategy for winning this game, and then convert that strategy into a formal proof that the triply quantified statement is true.