Days 1 + 2
We have a crumpled piece of paper on a pad with no overhang. What is our claim?
There exists a point, P, in the crumpled up paper that lies directly above where it started.
What is the symbol used to represent a union of two sets?
The “U” symbol is used
What does arbitrarily close mean in mathematical terms?
∀ ε > 0 ∃ Pn s.t. dist (Q, Pn) < ε
How to mathmaticate™ ‘Compactness' in ℝ or ℝ²?
A set S is compact if every sequence (an) has convergent subsequence.
What is the symbol used for representing the empty set?
∅
What are the things that could go wrong with the paper-pad situation and what C do they correspond with?
Ripping - continuity, infinitely large pad - compact, holes - complete, more than one pad - connected
What are the three properties of equivalence relations?
The three properties are reflexive, symmetric, and transitive.
How is the distance between two points in ℝ^n defined?
The distance between two points a(a1, a2, …, an) and b(b1, b2, …, bn) is defined as sqrt((a1-b1)^2 + (a2-b2)^2 + … + (an-bn)^2)
What is an interval?
I ⊆ ℝ is called an interval p.t. when every x,y ∈ I, y ∈ ℝ and x⊆y⊆z, then y ∈ I
What does ℕ represent and how is it defined(built)?
This represents the set of natural numbers.
1 ∈ ℕ, n ∈ ℕ implies n+1 ∈ ℕ
there is no such number k ∈ ℕ that k+1 = 1
What is the most important rule about sets?
An element can either be in a set or not in a set, it cannot be both
What is the Brouwer fixed point theorem?
If f: Ĩ^2 → Ĩ^2 is continuous, then P → f(P). f has a fixed point ∃ Q ∈ Ĩ^2 s.t. fQ) = Q
What does it mean for a sequence to converge to a point?
(An) converges to A p.t. ∀ε > O ∃ N ∈ ℕ s.t. ∀ n > N, dist(An, A) < ε
What does closed mean?
X ⊆ ℝ (or ℝ²) is said to be closed p.t. ∀ Cauchy sequences (xn) in X we have that (xn) converges to a point in X
What does ℤ represent and how is it defined?
This represents the set of integers.
ℤ = {[(a, b)]ℕ² | a, b ∈ ℕ, (a, b) ~ (c, d) means (a + d) = (b + c)}
How can we Mathmaticate™ the pad?
Ĩ^2 = [-1,1] x [-1,1]
What does it mean for a function to be bijective?
The function is both injective and surjective. Injective functions are described as “one-to-one”. Surjective functions are described as “onto”.
What makes a given sequence to be Cauchy?
The sequence (an) is Cauchy when for any ε > 0, there exists N s.t. for any n, m > N, distance between an, am is less than ε.
How can we say that the composition of two continuous functions is also continuous?
For two continuous functions F : X -> Y, G : Y -> Z, if we take any convergence sequence (xn) ⊆ X, which converges to x, as F is continuous (F(xn)) converges to F(x), and as G is continuous (G(F(xn))) converges to G(F(x)), which means that G∘F is also continuous.
What does ℚ represent and how is it defined?
This represents the set of rational numbers.
ℚ = {[(a/b)]ℤ*ℕ | a ∈ℤ, b ∈ ℕ, (a/b) ~ (c/d) means ad = bc}.
What is a function in mathematical notation?
F ⊆ S x T s.t. ∀ s ∈ S ∃! t ∈ T s.t. (s,t) ∈ F.
What does it mean for a function to be continuous?
F is continuous at a point p p.t. for any sequence (pn) s.t. pn -> p we have f(pn) -> f(p), F is continuous if F is continuous at p for all p ∈ Ĩ^2
Why did we introduce a new term, Cauchy sequence, when we can just simply say convergence?
There are some sequences that converge to a non-rational number, but the elements of the sequence are all rational numbers.
What is uniform continuity?
A function g: X -> Y (X,Y ∈ ℝ² ) is uniformly continuous p.t. ∀ ε > 0, ∃ δ>0 s.t. ∀ x1, x2 ∈ x, if dist (x1, x2) < δ, then dist(g(x1), g(x2)) < ε
What does ℝ represent and how is it defined?
This represents the set of real numbers.
ℝ = {[(an)] | (an) ⊆ ℚ, (an) ~ (bn) means (an-bn) -> 0)}.