Exams: Defs and Thms Flashcards
Closed Set in Rn
A set S⊆Rn is closed if Rn-S (the complement) is open
Adherent Point
Let S⊆Rn and x∈S. Then x is an adherent point if ∀r∈R+, B(x,r)∩S ≠∅
Accumulation Point
Let S⊆Rn and x∈Rn. Then x is an accumulation point of s if ∀r∈R+, B(x,r)∩(S{x})≠∅
Bounded in Rn
Let S⊆Rn. Then S is bounded if ∃x∈Rn and r>0 s.t. S⊆B(x,r)
Bolzano-Weierstrass Thereom
Every infinite bounded subset of Rn has an accumulation point.
Cantor’s Intersection Theorem
Let {Qi⊆Rn:i∈R+}:
1) Qi ≠ ∅
2) Qi is closed
3) Qi is bounded
4) Qi+1 ≤ Qi (subscript)
Then ∩(i=1->infinity)Qi ≠∅
Covering
Let S⊆Rn, Let F be a family of sets in Rn. We say F is a covering of S if ∪(A∈F)A⊇S:
-If ∀A∈F, A is an open cover
-If F is a countable collection then F is a countable cover
-If F is finite then F is a finite cover
-If F and G are covers of S and G⊆F then G is a subcover
Lindeloff Covering Theorem
Let A⊆Rn and F is an open cover of A. Then there exists a countable subcover G of A
Heine-Borel Theorem
If S⊆Rn which is closed and bounded and F is an open cover of S, then F has a finite subcover (closed and bounded subsets of Rn are compact)
Compact
Let S⊆Rn. We say S is compact if every open cover F has a finite subcover. (There exists G⊆F s.t. G is a finite and G covers S)
Metric Space
A metric space is a pair (M,d) where M is a set and d:MxM -> R is called a metric (or distance) such that∀x,y,z∈M:
1) d(x,x) = 0
2) d(x,y) > 0 if x≠y
3) d(x,y) = d(y,x)
4) d(x,y) + d(y,z) ≥ d(x,z)
Open Ball in Metric Spaces
Let (M,d) be a metric space and x∈M. The open ball centered at x of radius r>0 is B(x,r)={y∈M:d(x,y)<r}
Interior Point
If S⊆M then x∈S is an interior point if ∃r>o s.t. B(x;r)⊆S
Open Set in Metric Spaces
A set S is open if every point of S is an interior point
Closed Set in Metric Spaces
A set S is closed if M\S (complement) is open
Bounded in Metric Spaces
We say S is bounded if for x∈M ∃r>0 s.t. S⊆B(x;r)
Compact is Metric Spaces
Let (M,d) be a metric space and S⊆M. The set S is compact if every open cover has a finite subcover
Boundary
Let S⊆M, then the boundary of S is ∂ = S\∩(M-S)\ (put line over the elements to represent complement)
Convergence
Let (S,d) be a metric space and {xn} a sequence of points in S. We say that this sequence converges to a point p∈S if ∀Ɛ>0, ∃N∈Z+ s.t. if n≥N then d(xn,p)<Ɛ. We write xn -> p
Cauchy Sequence
A sequence {xn} in S is a Cauchy Sequence if ∀Ɛ>0 ∃N∈Z s.t. if m,n≥N then d(xm,xn)<Ɛ
Cauchy Complete
The metric space (S,ds) is Cauchy complete if every Cauchy Sequence is convergent
Limit
Let A⊆S and f:A->T a function if p∈T then limx->p f(x)=b if ∀Ɛ>0 ∃δ>0 s.t if 0<ds(x,p)<S then dt(f(x),b)<Ɛ. We write f(x)->b as x->p
Continuous (Vector Valued Functions)
Let f:S->T Be a function and p∈S then f is continuous at p if∀Ɛ>0 ∃δ>0 s.t if ds(x,p)<δ the dt(f(x),f(p)<Ɛ. We say f is continuous on A⊆S if f is continuous at every point in A
Triangle Inequality
or all x, y ∈ R we have
|x + y| ≤ |x| + |y|
Open Cover in Rn
Let S⊆Rn. An open cover of S is a collection C of open sets such that S C. The collection C of open sets is said to cover the set S
Field (And Field Axioms)
R is a field meaning R is equipped with two binary operations: +, * ∀ a,b,c∈R:
1) a+b=b+a, ab=ba (commutativity)
2) a+(b+c)=(a+b)+c,a(bc)=(ab)c (associativity)
3) ∃0,1∈R s.t. 0≠1 and a+0=a,a1=a
4) ∀ a∈R s.t. a≠0 ∃-a∈R s.t. a+(-a)=0 and ∃a^(-1)∈R s.t. a(a^(-1))=1
4) a(b+c)=ab+ac (distributivity)
Ordered Field (And Ordered Field Axioms)
An ordered field F is a field equipped with a binary relation < s.t. ∀ a,b,c∈F:
1) a>b,b<a, or a=b
2) If a<b and b<c then a<c (transitivity)
3) If a<b then a+c<b+c
4) If 0<a and 0<b then 0<ab
Upperbound/Lowerbound
Let S⊆R and x∈R then we say x is an upperbound for S if for all s∈S, s ≤ x (x is to the right of s). Similarly for lowerbound, s ≥ x (x is to the left of s)
Least Upperbound (supremum)/Most Lowerbound (infemum)
Let S⊆R and x∈R then we say x is a least upperbound for S if x is an upperbound for S and for every upperbound y for S, we have x≤y (smallest upperbound). Similarly for greatest lowerbound, x ≥ y
Complete Ordered Field
An ordered field F is (Dedekind) complete if every S⊆F which has an upperbound has a least upperbound. R is the unique complete ordered field.
Inductive
A set S⊆R is inductive if:
a) 1∈S
b) If x∈S then x + 1∈S
Fundamental Theorem of Arithmetic
Let n > 1 be an integer. Then n can be represented as a product of prime numbers in a unique way, apart from the order of the factors.
Dedekind Completeness Axiom
For any S⊆R s.t. S has an upperbound, S has a least upperbound.
Approximation Property
Let S⊆R where S≠∅ with b=sup(s). For every a∈R s.t. a < b there exists some x∈S s.t. a<x≤b
Set and Elements of a Set
A set is a collection of objects. The objects in a set are called elements.
Cartesian Products of Sets X and Y
X x Y = {(x,y) : x∈X, y∈Y}
Comparison Property
Suppose that S,T⊆R s.t. ∀s∈S, ∀t∈T. We have s≤t. Suppose S and T are bounded from above then sup(s)≤sup(t)
Function
A relation f⊆X x Y is a function if:
-∀x∈X ∃ y∈Y s.t. (x,y)∈f (f(x)=y)
-∀x∈X and ∀y1,y2∈Y if. (x,y1)∈f and (x,y2)∈f that implies y1=y2
Surjective (onto)
A function is surjective if f(x) = y (codomain(f) = range(f))
Injective (one-to-one)
A function is injective if ∀x1,x2∈X and x1≠x2 then f(x1)≠f(x2)
The Archimedean Property of R
If x,y∈R and x>0 then there exists a positive integer n∈Z+ s.t. nx>y
Bijective
A function that is both injective and surjective
Composition of functions
Let f : X → Y and g : Y → Z be functions. The composition of g and f , written g ◦ f , is the function of type X → Z given by (g ◦ f)(x) := g(f(x)).
Sequence
Let f : Z+ → X for some X {f(1), f(2),…} is a sequence and we write f(1) = fi so {f1, f2,…..} = {fn}
Equinumerous
Let X and Y be sets. We say X and Y are equinumerous (or similar) id there exists some function f: X → Y which is a bijection. We write X~Y
Finite
We say a set is finite if X~{1,…,n} for some n in Z+
Countably Infinite
A set X is countably infinite if X~Z+
Order Preserving
Let K: Z+ → Z+ we say that K is order preserving (or increasing) if ∀m,n∈Z+ if m<n then K(m)<K(n)
Subsequence
If {fn} is a sequence and K: Z+ →Z+ is order preserving then f◦K is a subsequence and is denoted {fK(n)}
Complement
The complement of a set in mathematics is the collection of elements not in the set.
Closed Set
A set S⊆Rn is closed if Rn complement S is open
Representational Theorem of Open Sets in R
Every open set in R is a countable union of disjoint open intervals.
Standard Basis Vector
ui∈Rn where ui=(0,…,0,1,0,….,0) (where 1 is the ith coordinate_ then {ui : 1≤i≤n} are the standard basis vectors
Open Ball in Rn
Let a∈Rn and let r∈R+ then B(a;r):{x∈Rn : ||x-a||<r} is the open ball centered a of radius r.
Component Interval
Let I be an open set in Rn. We say that I is a component interval of S if I is an open interval and there exists no open J⊆S s.t I⊊J
S ¯ (line over top)
{adherent points of S}. Closure of S
S’
{accumulation points of S}. Derived Set of S
Disconnected/Connected
A metric space (S,d) is disconnected if ∃ A,B∈S s.t. A∩B=∅, A,B≠∅, A∪B=S, and A,B are open. S is connected if it is not disconnected
Two-Valued Function
A two-valued function of f on D is a continuous function f:S→{0,1}
Components
Let F be the collection of containment wise maximal connected subsets of S. Then the elements of F are called components of S and any two components are disjoint.
Disjoint
A and B are disjoint sets if and only if A ∩ B = ∅
Arcwise Connected
Let S⊆Rn then S is arcwise connected if ∀a,b∈S, ∃ f:[0,1]→S which is continuous and f(0)=a and f(1)=b
