## Cartesian product

We know we can use binary operations to add two numbers, x and y: $x+y, x-y, x \times y, x \div y.$ Furthermore there are other operations such as $\sqrt{x}$ or any other root and exponents. Operations can involve other mathematical objects other than numbers, such as sets.

$def^n$ Given two sets, A and B, we can define multiplication of these two sets as the Cartesian product. The new set is defined as

$A \times B = \{(a,b): a \in A, b \in B\}$

Before looking at abstract examples, consider this case:

e.g.1. Assume there is a student in a self-catering residence and they want to make food preps for the first four days in the week. They want to know how many possible combinations they can make using fruits (between grapes and apples) and meals (pasta and meatballs, chicken wrap).

To solve this, let  $A = \{ \text{ grapes, apples } \} \text{ and } B = \{ \text{pasta and meatballs, chicken wrap} \}$

Then the possible meal options are: (grapes, pasta and meatballs), (grapes, chicken wrap), (apples, pasta and meat balls) and (apples, chicken wrap).

The Cartesian Product of sets A and B would be:

$\text{A x B} = \{( \text{ grapes, pasta and meatballs}), (\text{ grapes, chicken wrap }), (\text{ apples, pasta and meat balls }), (\text{ apples, chicken wrap}) \}$

We can think of the above example in more abstract terms.…

## 3. power sets

Recall powers (or exponents) of numbers: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

Similarly, sets have the power operation to create new sets.

$def^n$ If A is a set, then the power set of A is another set denoted as

$\mathbb{ P }(A) = \text{ set of all subsets of A } = \{ x: x \subseteq A \}$

Recall: A is a subset of B if every element in A is also in B. Furthermore, if A is a finite set with n-elements, then we can find the number of subsets in A by using this formula:

$2^n$

To find the power set of A, we write a list of all the subsets of A first – remembering that:

• the empty set is a subset of every set,
• and every set is a subset of itself

Let’s look at some examples:

e.g.1. $A = \{1, 2, 3 \}$

Using the formula $2^n$, we know that there are $2^3 = 8$ possible subsets of A, namely:

$\varnothing, \{1, 2, 3 \}, \{1 \}, \{2 \}, \{3 \}, \{1, 2 \}, \{2, 3 \} \text{ and } \{1, 3 \}$

Hence the power set is the set that contains all the above subsets:

$\mathbb{ P }(A) = \{ \varnothing, \{1, 2, 3 \}, \{1 \}, \{2 \}, \{3 \}, \{1, 2 \}, \{2, 3 \}, \{1, 3 \} \}$

Note: The cardinality (size) of  $\mathbb{ P }(A) = 8 = 2^3$ where size of A= 3 elements

e.g.2.

## 2. Subsets

https://giphy.com/gifs/infinite-boxes-vG1Dgq3JRXLMc

Consider a set $A = \{2, 3, 7\} \text{ and } B = \{2, 3, 4, 5, 6, 7\}.$ Note that every element in set A is also found in set B, however, the reverse is not true (B contains elements 4, 5 and 6 which are not in A)

Consider another case, $A = \{2n: n \in \mathbb{ N }\} = \{ 0, 2, 4, 6, ... \} \text { and } B = \mathbb{Z} = \{ ..., -2, -1, 0, 1, 2, ... \}.$ Again, we can see that every element in set A is also found in set B and similarly, everything in B cannot be found in set A. B contains negative and odd integers, which are not in A.

To describe this phenomena, mathematicians defined subsets:

$def^n$ Suppose A and B are sets. If every element in A is an element of B, then A is a subset of B and we denote this as $A \subseteq B$

If B is not a subset of A, as in the above cases, then there exists at least one element, say $x \in B \text{ such that } x \notin A. \text{ We denote this as } B \subsetneq A$

e.g.1. $\{2, 3, 5, 7, ... \} \subseteq \mathbb{ N }$ but $\{\frac{1}{3}, 2, 5, 7, ... \} \subsetneq \mathbb{ N }$ since $\frac{1}{3} \in \mathbb{ Q }$

e.g.2. $\mathbb { N } \subseteq \mathbb { Z } \subseteq \mathbb{ Q } \subseteq \mathbb{ R }$

e.g.3. $(\mathbb{ R } \times \mathbb{ N }) \subseteq (\mathbb{ R } \times \mathbb{ R })$ since  $(\mathbb{ R } \times \mathbb{ N }) = \{(x, y): x \in \mathbb{ R }, y \in \mathbb{ N }\}$ and $( \mathbb{ R } \times \mathbb{ R }) = \{ (x, y): x \in \mathbb{ R }, y \in \mathbb{ R } \}$ Hint: look at what sets y is in

Every set is a subset of itself :

e.g.1.

## Reverse Mathematics – By John Stillwell, a review

NB. I was sent this book as a review copy.

I’m not sure I’ve read a mathematics book which was so hard to review, not because of the quality of the book (which is superb), but because the way of thinking is in some senses so different to the way we normally think about mathematics. This, indeed, is also the book’s best feature. This book gets you thinking about mathematics in ways which I have never explored before, and which have definitely given me a new, and I think, improved perspective on formal mathematics.

In general in mathematics we start with a set of assumptions (axioms), and explore the consequences of them. Within Euclidean geometry we start with ideas about lines, and points, and circles, and then see what other theorems can be proved from these. Within set theory too, we start with a set of ideas about equalities of sets, existence, pairings, unions etc which we hold to be true and then see what can be said of other properties of sets, which are not straightforwardly stated in the axioms.…

## Singalakha’s guide to plotting rational functions

To sketch the graph of a function $k(x)=\frac{f(x)}{g(x)}$:

1. Find the intercepts:
1. X-intercepts, set y=0 (there can be multiple)
2. Y-intercept, set x=0 (there can be only one)
2. Factorise the numerator and denominator if possible:
1. Sign table: determine where the function is negative and where it is positive
3. Find the Vertical asymptotes:
1. This occur if the function in the denominator is equal to zero, i.e $g(x) = 0$, AND that in the numerator must not be zero, i.e $f(x)\ne 0$.
4. Find any Horizontal asymptote:
1. If the degree of the function in the numerator, i.e $f(x)$, is less than the degree of the function in the denominator, i.e $g(x)$, then the horizontal asymptote is the line $y = 0$.
2. If the degree of the function in the numerator, i.e $f(x)$, is equal to the degree of the function in the denominator, i.e $g(x)$, say for example, the degree of $f(x)$ and $g(x)$ is $n$ for some non-negative $n$ element of integers, then there is a horizontal asymptote.

## My vlogging channel

Hi all, I’m not sure if it counts as vlogging, or making maths videos regularly fits into a slightly more niche category, but anyway, I wanted to advertise some videos that I’ve been putting up recently. I’m doing this in an attempt to find a different communication channel with my first year maths class, and so far the videos are getting reasonable feedback. I have a long way to go in terms of making them slick, and I goof up from time to time, but it’s an interesting experience. If you have specific questions that you would like me to discuss in a video. Let me know.

In this video I talk about a method for solving inequalities involving absolute values:

 How clear is this post?

## 1. Sets

If like me, you’ve spent most of your mathematical high school years introduced to basic sets at the beginning of the year from Grades 8 to 12, then I think you’d agree that sets was one of the quickest and easiest sections we traditionally did. We would quickly recap the same fundamental properties of sets before moving onto more interesting topics, usually algebra. The section would go a little bit like this:

• define the differences between whole and natural numbers, integers, rational numbers and real numbers
• define the differences between unions, intersections and complements, usually through the understanding of Venn-diagrams
• use set builder notation (introducing algebra through this)

If like myself, you truly believed that this was as complicated as sets could ever get, then you, dear reader, like my former-myself, are in for a treat. In university, we build on these basic ideas and have a more in depth understanding about the importance of sets and their greater role in the scheme of mathematics.…

## A quick introduction to writing mathematics in WordPress using LaTeX

Here are a couple of very useful links about writing mathematics, for new authors of this blog:

I will update this as I find more useful material.

• Generally I like to use the Visual Tab on the editor here rather than the Text Tab, unless there is some sort of strange formatting in which case I will go in and alter the Text.
• I usually like to put formulas centrally justified on their own on a line with blank lines above and below.
• Add Media to upload pictures or gifs and use the Fusion Shortcodes button (to the left of the yellow star in the blue box), to embed Youtube content.

Please let me know if, as an author, there is anything which is unclear about posting here and I will update accordingly.…

## e-day – A mathematical holiday celebrated on February 7th

Today, February 7th, 2018, is called e-day because e is approximately 2.718, and this date is written 2/7/18 in some parts of the world.

e, also called Euler’s Number after the Swiss mathematician Leonhard Euler, is a very important constant that comes up in many different places in mathematics. The numer e was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest where e arises as the limit of (1 + 1/n)n as n approaches infinity. e can also be calculated by summing:

The constant e appears naturally on the exponential function, which models growth. Hence, the same way that the constant π appears in everything that is round, the number e appears in everything that grows: size of baby animals, leaves in trees, bacteria populations, spreading of diseases, spirals in flowers and snails, radiactive decay of elements, money invested in a bank, processing power of computers… Everything that grows the faster the bigger it is follows an exponential law, and contains the number e.

## Can you find a simple proof for this statement?

I thought more about the last question I added into the addendum of the Numberphile, Graph theory and Mathematica post

It can be succinctly stated as:

$(\forall m\in\mathbb{Z}, m\ge 19) (\exists p,q\in\mathbb{Z}, 1\le p,q such that $\sqrt{p+m}\in\mathbb{Z}$ and $\sqrt{q+m}\in\mathbb{Z}$.

In words:

For all integers m, greater than 19, there are two other distinct positive integers less than m such that the sum of each with m, when square rooted is an integer.

What is the shortest proof you can find for this statement?

 How clear is this post?