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## 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:

• 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.

## 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. …

## 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.…

## 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.…

## 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.…

## 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!…

## 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.…

## 0.4. 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.…

## 0.3. 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.