S2, C1: Orbits, Functions and Symmetries

Question 1

What are the symmetries of a square?

Answer 1

r0: rot0
r1: rot pi/2
r2: rot pi
r3: rot 3pi/2
s1: ref 0
s2: ref pi/2
s3: ref pi
s4: ref 3pi/2
Note: rotations are taken to be anticlockwise.

Question 2

What is a function, domian, codomain, range and image?

Answer 2

A function f: A -> B assigns, for each aϵA, a unique element f(a) of B. The set A is called the domain of f and B is the codomain of f. The range or image of f is the set of all things ‘hit’ by the function.

Question 3

Composite functions.

Answer 3

Let f : A → B and g : B → C be functions. The composite g ◦ f is the function A → C such that, for all a ∈ A, g ◦ f(a) = g(f(a)).

Question 4

Identity function.

Answer 4

The function from A to A which sends every element to itself is called the identity function on A.

Question 5

Inverse function.

Answer 5

If f: A -> B is a function, then an inverse for f is a function g: B -> A s.t. g ◦ f = id and f ◦ g = id.

Question 6

Using polar coordinates, we define two functions rotφ : R^2 → R^2 and refφ : R^2 → R^2 by…

Answer 6

rotφ((r, θ)) = (r, φ + θ)

refφ((r, θ)) = (r, φ − θ)

Question 7

Composing rotations and reflections.

Answer 7

rotα rotβ = rotα+β
refα refβ = rotα−β
rotα refβ = refα+β
refα rotβ = refα−β

Question 8

A function f:A -> B has an inverse iff…

Answer 8

it is bijective

Question 9

A set is countable iff…

Answer 9

there is a bijection f: N -> A (N is natural numbers)

i.e. countable sets are the same size as the set of natural numbers

Question 10

Drawing a shape with a group of symmetries of order p.

Answer 10

Draw a regular p-gon and add L shaped lines to vertices..

Question 11

Using the Chinese remainder theorem to solve simultaneous congruences.

Answer 11

x (congruent) a (mod b) and x (congruent) c (mod d).
Then find a solution to: bs + dt = 1.
Thenx is congruent to bsd + dtb (mod bd).