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## 1.2 Properties of Groups

Recall the definition of a group:

A set G is “upgraded” into a group if it satisfied the following axioms under one binary operation (*) :

1. Closure: $\forall x, y \in G, x*y \in G$
2. Associativity: $\forall x, y, z \in G, (x*y)*z = x*(y*z)$
3.  Identity: $\exists e \in G, \text{ called the identity element such that } \forall x \in G, x*e = e*x = x$
4. Inverse:  $\exists y \in G, \text{ called the inverse of x, with } x*y = y*x = e \forall x \in G$

An Abelian group is a group that is follows the axioms 1 – 4 with the addition of one property:

1. Commutativity: $\forall x, y \in G, x*y = y*x$

In addition to the axioms, the following properties of groups are important to note:

1. Uniqueness of the identity element
2. Uniqueness of the inverse element
3. Cancellation law
4. Inverse property (extended)

Uniqueness of an element in mathematics means there exists only one such element with that property. We prove uniqueness by making an assumption that there are two elements in the set that satisfy the property, and show that if such a situation holds, then the two elements must be equal!

We use * to denote the binary operation between elements and “QED” to signal the end of the proof.

The remainder of the post aims to go through the proofs of these properties!…

## 1.1 Groups Introduction

Binary operations are operations such as addition, subtraction, multiplication, division, modulus etc. that are applied to two quantities.

example 1: $2+5$ is an example of an expression with addition as the binary operation

example 2: Let f and g be functions defined on sets A to B. Then the composition of the functions $\text{ f(g(x)) }$ is a binary operation

We will use * to denote an arbitrary (general) binary operation.

A set G is “upgraded” into a group if it satisfied the following axioms under one binary operation (*) :

1. Closure: $\forall x, y \in G, x*y \in G$
2. Associativity: $\forall x, y, z \in G, (x*y)*z = x*(y*z)$
3.  Identity: $\exists e \in G, \text{ called the identity element such that } \forall x \in G, x*e = e*x = x$
4. Inverse:  $\exists y \in G, \text{is called an inverse element of } x \in G \text{ with } x*y = y*x = e$

An Abelian group is a group that is follows the axioms 1 – 4 with the addition of one property:

1. Commutativity: $\forall x, y \in G, x*y = y*x$

For the remainder of this post, we will explore these axioms and look at some examples

Closure: $\forall x, y \in G, x*y \in G$

This means we can take any elements in the set G and perform the operation defined by * and the result will also be an element in the group.…

## 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.

## 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.…