Structures: Maths 1st Partial

Given two sets A and B, we say that A is subset of B if:

_{^{(Definition 1)}}

A ⊆ B if all the elements of A are also elements of B

aka if x ∈ A implies x ∈ B

What’s another way of saying: When both A ⊆ B and B ⊆ A, and what does it mean

x ∈ A if and only if x ∈ B

it means A = B

What is it called when A ⊆ B and A ≠ B

A is a proper subset of B

What’s is called when the set A = {a}

A singleton

Given two sets A and B, their intersection A ∩ B is:

_{^{(Definition 2)}}

The set of all elements that belong to both A and B

aka x ∈ A ∩ B if x ∈ A and x ∈ B

What is the empty set?

_{^{(Definition 3)}}

The set without elements: ∅

It is a subset of any set

When two sets A and B have A ∩ B = ∅:

they are disjoint

When A ∩ B = A, what does this tell us about the relationship between A and B

A ∩ B = A if and only if A ⊆ B

How to do a proof of an ‘if and only if’

A=1 if and only if B=2

Proof:

“If”. Let B=2. We want to show that A=1. Here there is some rule that will function e.g. to prove that two sets are equal, prove that they are both subsets of each other.

We have shown that both __ and __ hold; we can therefore conclude that A=1, which completes the proof of the “if” part

“Only if”. Let A=1. We want to prove that B=2. etc etc

Given two sets A and B, their union A ∪ B is:

_{^{(Definition 5)}}

The set of all elements that belong to A or B (or both)

aka x ∈ A ∪ B if x ∈ A or x ∈ B

Given two sets A and B, their difference A - B is:

_{^{(Definition 6)}}

The set of all elements that belong to A but not to B

aka x ∈ A \ B if both x ∈ A and x ∉ B

Talk about the complement set:

The general set of reference is called universal set / space. Let’s denote it as S. Given any of its subsets A, the difference S - A is denoted by A^c and is called the complement set (/ the complement) of A.

The difference operation is called complementation when it involves the universal set.

The operations of union and intersection are both:

• Commutative

• Associative

What does commutative mean?

For any two sets A and B, we have A ∩ B = B ∩ A and A ∪ B = B ∪ A

What does associative mean?

For any three sets A, B and C, we have A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C

When only finitely many of the sets A₁, A₂, …, A_n, are distinct and the rest are just repetitions, what happens to the infinite union and intersection?

They reduce to the finite case

What does it mean that the operations of union and intersection are distributive?

Given any three sets A, B and C we have: ∪ ∩

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

and

A u ( B n C) = (A u B) n (A u C)

When is a family {A₁, A₂, …, A_n} of non-empty subsets of A a partition of A?

When the subsets are pairwise disjoint

aka when A_i n A_j = ∅ for every i ≠ j

e.g. A = {1,2,3,4,5}, The sets A₁ = {1,2} and A₂ = {3,4,5} form a partition of A.

What does a closed bounded interval look like?

_{^{(intervals in R, 1.4-1.5)}}

[a,b] = {x ∈ ℝ: a ≤ x ≤ b}

What does an open bounded interval look like?

(a,b) = {x ∈ ℝ: a < x < b}

What does a half-closed bounded interval look like?

(a,b] = {x ∈ ℝ: a < x ≤ b}

What does an unbounded interval look like?

[a,∞) = {x ∈ ℝ: x ≥ a}

What is the complement of the unbounded interval?

(-∞, a] = (a, ∞)^c

The bracket for infinity is always (, but when doing the complement the bracket for ‘a’ switches from ] to ( or vice versa

What are the endpoints of these intervals?

a and/or b. Even if its only a < sign.

A subset I of ℝ is an interval if:

_{^{(Definition 30)}}

given any two points x and y of I with x ≤ y, all points z ∈ ℝ such that x ≤ z ≤ y belong to I

Does the upper bound have to be the smallest upper bound?

_{^{— ok before i was doing up to textbook page idk but flashcard 18 was 35 but now im doing from my goodnotes notebook page 7}}

No - any number above the range is an upper bound

What do we mean by A* and A_*

A* = the set of all upper bounds

A_* = the set of all lower bounds

A is unbounded if A* and A_*..

A* ≠ ∅ or A_* ≠ ∅

How do we define an element M ∈ A as a maximum of A

if M ≥ x ∀ x ∈ A

What do we call the smallest upper bound (Min A*)?

Sup A

What do we call the greatest lower bound (Max A_*)?

Inf A

What separates sup A from the maximum of A

Maximum of A has to be an element of the set, sup A does not

e.g. A is the set of x = 1/m, where m is in the set of natural numbers. as m increases, A gets closer and closer to zero. Zero is inf A but not min A as it isnt part of the set but is defo the biggest lower bound.

What is the least upper bound principle?

Each non-empty set A ⊆ R has supremum if it is bounded above and infimum if it is bounded below

If A is not bounded above what is Sup A?

∞

What does a = Sup A mean for a?

How can |x| be thought of as a distance

because |x - 0| = d(x, 0)

Define a neighbourhood

An open interval such that X₀ is the mid point

How can we calculate the inner product (scalar product) of two vectors x̲ and y

x • y = x₁y₁ + x₂y_{2 +} x₃y₃ + …

What is a condition for doing this

x,y ∈ ℝ^m , aka x and y must have the same number of components

What’s it called when x • y = 0

x is orthogonal to y (x ⟂ y)

What are the three types of ‘greater than’

x ≥ y - (weakly) greater than

x > y - strictly greater than

x >> y - strongly greater than

what do the three types of ‘greater than’ signify?

Weakly greater than: x_i ≥ y_i for every i from one to m - aka the two vectors could even be equal

strictly greater than: x_i ≥ y_i for every i from one to m, but x_j > y_j at least for one j

strongly greater than: x_i > y_i for every i from 1 to m

Define convexity mathematically

How do we find ||x||

sqrt(x•x)

What do standard unit vectors (or versors) look like in ℝ² and ℝ³

ℝ² : e₁ = (1,0) e₂ = (0,1)

ℝ³ : e₁ = (1,0,0) e₂ = (0,1,0) e₃ = (0,0,1)

there can only be one 1 so the position of it within the 0s is what we track

What is normalising a vector?

Dividing it by it’s own length to rescale x to length 1 without changing its direction

What is the formula for normalising a vector x ≠ 0

. x

v = —

. ||x||

How can we normalise x = (1,1,1)

||x|| = sqrt (1+1+1) = sqrt (3)

so v = (1/sqrt(3), 1/sqrt(3), 1/sqrt(3))

What is the triangle inequality

(given that x,y are vectors that are elements of the set of real numbers for an undefined m of dimensions)

||x + y|| ≤ ||x|| + ||y||

_{^{aka pythagoras theorem, going from point A to C can never be longer than going from A to B then C}}

How can we write the distance of two vectors d(x,y)

in ℝ: |x-y|

in ℝ^m: ||x-y||

what does it mean about x and y when d(x,y) = 0

x = y

How do you mathematically define a point x₀ being interior for A

if ∃ Ɛ > 0: B_Ɛ(x₀) ⊆ A

there exists a value for sigma (the radius) where for every point in this neighbourhood with a tiny radius, the little circle neighbourhood is copmletely a subset of A and therefore the midpoint (x₀) is interior to A

How do you mathematically define a point x₀ being exterior for A

∃ Ɛ > 0: B_Ɛ(x₀) ⊆ A^C

there exists a sigma greater than zero such that the neighbourhood circle around x₀ is fully a subset of A complement

How do you mathematically define a point x₀ being a boundary point for A

∀ ε > 0, B_ε(x₀) ∩ A ≠ ∅ and B_ε(x₀) ∩ Aᶜ ≠ ∅

for every radius of the neighbourhood of the circle around x0, the intersections of the circle with A and A^c are not empty - no matter how small the circle, a little is in A and A^c

If x₀ is a boundary point for A if and only if…

it is also a boundary point for A^C

What is the notation for the set of interior points to A

Int A

Notation for the set of boundary points of A (frontier)

𝜕 A

Notation for the set of exterior points

Int A^C

Notation for the closure of A

(if A is an open set, this means it includes the interior points of A U the set of boundary points of A)

Ā

How do you mathematically define a point x₀ being an isolated point

∃ Ɛ > 0 such that B_Ɛ(x₀) n A = {x₀}

When A is a skinless circle (open set), all points of A are

All points in A are interior

A ⊆ ℝ^m is closed if..

all boundary points belong to A

𝜕 A ⊆ A

What is ℝ extended

when ℝ = [-∞,∞] instead of (-∞,∞)

When A = [a, ∞), A is __ and the set of boundaries is __

A is closed (as it contains all its boundary points)

the set of boundaries is {a} - a is a singleton set of boundaries

How do you mathematically define a set A ⊆ ℝ^m as bounded

∃ K > 0 such that ‖x‖ < K ∀ x ∈ A

When is a set A ⊆ ℝ^m compact

If its closed and bounded

All neighbourhoods are __

open

A ⊆ ℝ^m is closed if and only if A^C

is open

When A = ∅ the complement set is..

A = (ℝ^m)^C

The intersection of a finite number of open sets is ___

open

The union of open sets is

open

The union of a finite number of closed sets is

closed

The intersection of closed sets is

closed

What is the definition of a limit point (point of accumulation)

Let A ⊆ ℝ^m

∀ ε > 0, ∃ x ∈ A, x ≠ x₀ such that x ∈ Bε(x₀)

However u choose the radius, there is an x that satisfies the set A, that is not the center point, however small the radius might be

Can an isolated point be a limit point?

no

the definition states when x does not equal x₀ and the only point of A in the neighbourhood around an isolated point is the point itself

Are these limit points:

• Interior

• Exterior

• Boundary point where x₀ ∉ A

• Boundary point where x₀ ∈ A

• Interior: yes

• Exterior: no

• Boundary point, ∉ A = not isolated : yes

• boundary point, ∈ A

  • not isolated: yes

  • isolated: no

How do we denote and name the set of limit points of A?

A’ - called the derived set

How do we denote the set of isolated points

A \ A’ = {isolated points}

What does it mean when A’ ⊆ A

A is closed