Cantor–Schröder–Bernstein Theorem

Knowledge this posts assumes: What is a set, set cardinality, a function, an image of a function and an injective (one-to-one) function.

David Hilbert imagines a hotel with an infinite number of rooms. In this hotel, each room can only be occupied by one guest, and each room is indeed occupied by exactly one guest. What happens if more guests show up? Can they be accommodated for?

PAUSE: WHAT DO YOU THINK AND WHY?

Suppose we propose they cannot be accommodated for, since all the rooms are occupied. Hilbert then claims that he can define the functions f:A \mapsto B, and g:B \mapsto C, where A is a set containing all current guests, and f simply maps each guest to a room in the set B, and g maps each room in B to a new one in C. Notice that these functions must be injective, since if a room contains two different guests, those two different guests must be the same guest; recall f(a) = f(b) \rightarrow a = b.…

By | August 16th, 2019|Uncategorized|0 Comments

1.6 Partitions

Recall the  relation \equiv \text{ mod} (4) on the set \mathbb{ N}.

One of the equivalence classes is [0] = \{ ..., -8, -4, 0, 4, 8, ...\} which is equivalent to writing [0] = [4] = [-4] = [8] = [-8] ...

We could do this because the equivalence class collects all the natural numbers that are related to zero under the relation \equiv \text{ mod} (4)

 

The following theorem generalises this idea for any relation \equiv \text{ mod} (n) on the set \mathbb{ N}: for the integer n.

Let R be an equivalence relation on set A. If a, b \in A,  then [a] = [b] \iff  aRb.

Essentially, equivalence classes  [a] = [b] are equal if the elements  a, b \in A, are related under the relation R. And simultaneously, knowing that elements a, b \in A, are related under R means their equivalence classes  [a] = [b] are equal.

An equivalence class  \equiv \text{ mod} (n) divides set a A into n equivalence classes. We call this situation a partition of set A.

A partition of a set A is defined as a set of non-empty subsets of A, such that both these conditions are simultaneously satisfied:

 (i) the union of all these subsets equals A.

(ii) the intersection of any two different subsets is

 

Let’s return to our example: \equiv \text{ mod} (4) on the set \mathbb{ N}. We could represent this set as:

Modulus 4, General

  • NOTE: Each equivalence class above represents an infinite set and despite the drawing suggesting [0] is larger than [3] for instance, this is not true.
By | August 9th, 2019|Uncategorized|0 Comments

Review: Calculus Reordered

Book title: Calculus Reordered: A History of the Big Ideas
Author : David M. Bressoud

9780691181318

Princeton University Press
Link to the book: Calculus Reordered: A History of the Big Ideas

Discussions on the history of different fields are usually dry, wordy and generally, when you are studying the field, hard to read. This is because they are usually geared towards the general audience, and in doing so most authors tend to strip away the very exciting technical details. I expected the same treatment from the author, but I was pleasantly surprised.

The book contains 5 chapters, which are the following:

1) Accumulations
2) Ratios of Change
3) Sequences of Partial Sums
4) The Algebra of Inequalities
5) Analysis

Each of these chapters has a central theme that is being covered, but they are not at all disjoint. For instance, the last three contain the history of concepts that would normally be found in a first course for Real Analysis, while the first two are essentially the more applied spectrum to serve as some form of motivation for going through all this trouble, although they can certainly stand on their own.…

By | August 4th, 2019|Uncategorized|1 Comment

Investigating Practical Ordering of Grids

In Reinforcement Learning there is an environment known as Gridworld. In this environment you have a grid and there is an agent that learns how to find the shortest path from one cell to another. The theme of reinforcement learning is that you do not want to hard-code the rules, but you want the agent to explore until it can find a set of moves that are optimal for the problem at hand. Usually you can alter the grids to make the tasks tough–set ‘traps’, add obstacles, etc. We are considering grids with obstacles, and an interesting question that came up is the following,

Given two grids of size {N}, say {G, \,G'} which have respectively {k,l} obstacles where {k,l\in \mathbb{N},\,k,l\geq 0}, what are reasonable ways to put an order on the ‘complexity’ of the grids?

In other words, we want to be able to say that, for instance, in {G} the agent will find the optimal path more easily than in {G'} given any two grids {G,G'}.…

By | July 28th, 2019|Uncategorized|0 Comments

1.5 Equivalence classes (Infinite sets)

 

Let’s find the equivalence classes of the following finite set S:

Given S = \{ -1, 1, 2, 3, 4 \}, we can form the following relation R = \{ (-1, -1), (1,1), (2,2), (3,3), (4,4), (1,3), (3,1), (2,4), (4,2) \}.

Note: writing the relation R on set S in the following ways is equivalent:

-1R-1, 1R1, 2R2, 3R3, 4R4, 1R3, 3R1, 2R4, 4R2

or

-1\le -1, 1 \le1, 2 \le2, 3  \le3, 4 \le4, 1 \le 3, 3 \le 1, 2 \le 4, 4 \le 2

This relation, R has been given the symbol \le but it means “the same sign and parity” in this case. For instance, (1,3) or 1 \le 3 tells us that one and three are both odd and both have the same sign in set A (both positive).

The equivalence classes for this relation are the following sets:

\{ -1 \}, \{ 1, 3\} \text{ and } \{2, 4 \}

We obtained the above equivalence classes by asking ourselves:

  • How is the element -1 related to any other element in the set S under the definition of R?

Since R is defined as “the same sign and same parity,” then we’re really asking ourselves whether -1 has the same sign as any other element in S. Since all the other elements are positive, then -1 has the equivalence class containing only itself. Another question we would’ve asked ourselves is whether -1 is even or odd. …

By | July 25th, 2019|Uncategorized|1 Comment

On the invariant measure in special relativity

I’m writing this for my string theory class. We are basing our lectures on Zwiebach – A First Course in String Theory, and starting off with special relativity. Not everybody in the class has a physics background (pure and applied mathematics students), and so there are likely to be questions which come up which show where I have to fill in some knowledge. We had a question about the invariant measure in special relativity (SR) and why there was a different sign in front of the time term compared with the space terms. I’ll do my best to explain here. Note that I am not explaining it in the precise chronological order of discoveries.

We start the picture off with relativity before SR – that is, Galilean Relativity. This simply states that the laws of motion are the same in all inertial (non-accelerating frames). That may sound straightaway like SR, but there’s a crucial ingredient missing which we will see in a bit.…

By | July 23rd, 2019|Uncategorized|0 Comments

1.4 Equivalence classes

Let’s recall the definition of an equivalence relation:

 A relation R on a set A is termed an equivalence relation if it is simultaneously reflexive, symmetric and transitive.

Let’s look at more examples:

Example One: Let A = \{2, 11, 17, 20\} be a set with the following relation: R = \{ (2,2) (11,11) (17,17) (20,20) (2,20) (20,2) (11,17) (17,11) \}.

The relation described by R is termed “the same parity.” Elements x and y are said to have the same parity if they are both odd or both even. In our case, the elements 11 and 17 are both odd – hence have the same parity. Similarly, 20 and 2 have the same parity because they are both even. An element will always have the same parity as itself.

The elements that share the same parity as 11 can be grouped together to form a set: O = \{ 11, 17 \}. This is the set of all odd elements from A.

Similarly, the even elements can be grouped together to form the set: E = \{ 2, 20\}.

The new sets, O and E, form the equivalence classes of the relation R on set A.…

By | July 22nd, 2019|Uncategorized|0 Comments

1.3 Relations: Equivalence relations

We know that a relation is called an Equivalence Relation when it is reflexive, symmetric AND transitive on some set A. Let’s look at some examples.

Example One: Let a,b \in \mathbb{R}. Suppose we have that a is related to b (i.e. a ~ b) if a - b \in \mathbb{Z}. We want to show that our relation ~ is an equivalence relation.

First, let’s unpack what the question requires us to prove: It wants us to show that the relation ~ on set A is an equivalence relation. Hence, we need to show that ~ is reflexive, symmetric and transitive.

The relation ~ (in this case) is defined as follows: IF any two real numbers, a and b, are related THEN we know that a – b is some integer.

It’s important to note that the order in which a relation is important! Always write your equations as they’ve been given in the question to avoid confusion and mistakes :)

Ok, let’s prove this.…

By | July 18th, 2019|Uncategorized|1 Comment

1.2 Relations: Properties

Note: Do not confuse binary operations (+, x, -, …) with relations. Recall the definition for a relation as:

A relation R on a set A is a subset R ⊆ A × A. We often abbreviate the statement (x, y) ∈ R as xRy.

For instance, the binary operation “x” has a numeric value: 3 \times 3 = 9 \text{ and } 40 \times \frac{1}{5} = 8. Yet a mathematical relation, for example “<“, has a True/False value: 3 < 3 \text{ and } 40 < \frac{1}{5} are both False expressions.

 

We want to look at some properties of relations. We will look at three properties for relation expressions:

Suppose A is a set with relation R, then

  1. Relation R is Reflexive if \forall x \in A, \text{ } xRx
  2. Relation R is Symmetric if \forall x,y \in A, \text{ } xRy \rightarrow yRx
  3. Relation R is Transitive if \forall x,y,z \in A, \text{ } xRy \wedge yRz \rightarrow xRz

The first property tells us that if every element, x, in set A is related to itself, then the relation R acting on the set A is termed “reflective.”

The second property tells us that if “x is related to y” from set A implies that “y is also related to x,” then R is termed “symmetric.”

Lastly, if “x is related to y” and “y is related to z” implies that “x is also related to z,” then the relation R is termed “transitive.”

 

Let’s look at the following examples: A = \{ 1, 2, 3, 4\} with relation \le. Then:

\forall x \in A, \text{ } x \le x

In other words \le is reflexive since every number in set A is equal to itself (i.e.…

By | July 18th, 2019|Uncategorized|0 Comments

1.1 Relations: Introduction

What are relations?

In every day life, a person is related to their parents, siblings, cousins, teachers, friends, etc. in some way. Similarly in mathematics, mathematical objects like numbers and sets are related to one another in some way. Many relations (symbols) will be familiar already:

  1. 2 <3
  2. \pi \approx 3.14
  3. 5 \in \mathbb{Z}
  4. X \subset Y
  5. a \equiv b(modn)

Consider the following set A = \{1, 34, 56, 78 \}. We can compare the numbers in A using the symbol “<” as follows: 1 < 56, 34 < 78 etc. We can write this as a set in the following way: R = \{ (1,34), (1,56), (1,78), (34,56), (34,78), (56,78) \}.

Each pair in this new set R expresses the relationship x < y (where x and y are numbers from A).

In other words, 1<34, 1<56, 1<78, ... So if asked whether 34 < 78 is true,  one only needs to look into our set R to find the pair (34,78). If we didn’t find it, then the relation would be considered false for the given set. The above example is intuitive because we are already comfortable with the relation <. In more abstract cases, thinking of the relationship between mathematical objects in this way may be a little trickier!…

By | July 18th, 2019|Uncategorized|1 Comment