January 30, 2018 - Page 3 of 3 - Shawn Zhong

$Sets • Contains ○ If A is a set and x is an element of A (an object of A), then we write x∈A ○ Otherwise, we write x∉A • Set ○ The empty set or null set is a set with no elements, and is denoted as ∅ ○ If a set has at least one element, it is called nonempty • Subset ○ If A and B are sets and every element of A is an element of B ○ Then A is a subset of B ○ Rubin write this A⊂B, or B⊃A ○ A⊂A for all sets A • Proper subset ○ If B contain something not in A, then A is a proper subset of B • Equal ○ If A⊂B and B⊂A then A=B. ○ Otherwise A≠B √2 is Not Rational • We proved that √2 is not rational • i.e. there is no rational number p such that p^2=2 • Let A={p∈Qp^2<2}, B={p∈Qp^2>2} • Prove: A has no largest element, and B has no smallest element ○ Let p∈Q, and p>0 ○ Let q≔p−(p^2−2)/(p+2)=(2p+2)/(p+2),then q^2−2=((2p+2)/(p+2))^2−2=2(p^2−2)/(p+2)^2 ○ If p∈A § then p^2−2<0 § ⇒q^2−2=2(p^2−2)/(p+2)^2 <0 § ⇒q^2<2 § ⇒q∈A § ⇒q>p § i.e. A has no largest element ○ If p∈B § then p^2−2>0 § ⇒q^2−2=2(p^2−2)/(p+2)^2 >0 § ⇒q^2>2 § ⇒q∈B § ⇒q</p> <p § i.e. B has no smallest element What is R? • The real numbers are an example of field. • A field is a set F with two binary operations called addition and multiplication that satisfy that following axioms: • Axioms for addition (+) ○ (A1) If x∈F and y∈F, then x+y∈F ○ (A2) Addition is communicate: x+y=y+x,∀x,y∈F ○ (A3) Addition is associative: (x+y)+z=x+(y+z),∀x,y,z∈F ○ (A4) There exists 0∈F s.t. x+0=x, ∀x∈F ○ (A5) ∀x∈F, there exists an additive inverse −x∈F s.t. x+(−x)=0 • Axioms for multiplication (× or ⋅) ○ (M1) If x∈F and y∈F, then xy∈F ○ (M2) Addition is communicate: xy=yx,∀x,y∈F ○ (M3) Addition is associative: (xy)z=x(yz),∀x,y,z∈F ○ (M4) F contains an element 1≠0 s.t. 1⋅x=x, ∀x∈F ○ (M5) If x∈F and x≠0, then there exists 1/x∈F s.t. x⋅1/x=1 • (D) The distributive law: x(y+z)=xy+xz, ∀x,y,z∈F$

Math 521 – 1/24

$Course Overview • The real number system • Metric spaces and basic topology • Sequences and series • Continuity • Topics from differential and integral calculus Grading Homework assignments 20% Quiz (Feb. 9) 5% Midterm 1 (Mar. 9) 20% Midterm 2 (Apr. 13) 20% Final (May. 10 7:45-9:45 AM) 35% A ≥90% B ≥80% C ≥70% D ≥60% Tutoring • Tom Stone @VV B205 • Monday 2:30 - 4:30 PM • Tuesday 2:00 - 4:00 PM What is Analysis • Proof • How calculus works • Fundamental axioms Number Systems • Natural Numbers: N={1,2,3,…} • Integers: Z={0,±1,±2,±3,…} • Rational Numbers: Q={a/b│a,b∈Zb≠0} • Real numbers R: fill the "holes" in the rational numbers √2 is not rational • There is no rational number p such that p^2=2 • Proof by contradiction • Assume there is a rational number p such that p^2=2 • Then p=m/n , where m,n∈Z, n≠0, and m,n have no common factor • (m/n)^2=2⇒m^2/n^2 =2⇒m^2=2n^2 • So m is even • m=2k (k∈Z⇒(2k)^2=2n^2⇒4k^2=2n^2⇒2k^2=n^2 • So n is also even • m,n are both division by 2 • This contradicts the fact that m,n have no common factor • So no such p exists$

Shawn Zhong

Shawn Zhong

Math 521 – 1/29

Math 521 – 1/26

Math 521 – 1/24

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