15.1. Definition of Sets

A very high level definition is, it is an unordered collection of objects and what I mean by unordered collection of objects here is that ordering of the elements in the set does not matter. So, for instance, if I have a set consisting of the elements 1, 2 and 3 and then it does not matter whether I list them as 1, 2, 3 or whether I list them as 3, 2, 1 both will be the same sets.

A set is defined as a collection of distinct objects, where the order of these objects is irrelevant. For example, the set {1, 2, 3} is the same as the set {3, 2, 1}; they contain the same elements but arranged differently. This characteristic signifies that within the context of sets, what truly matters is the existence of an element rather than its position in a list.

Think of a set like a bag of different colored balls. If you have a bag containing red, blue, and yellow balls, having them in any order (red, blue, yellow or yellow, blue, red) still makes it the same bag. The uniqueness and the presence of colors in the bag matter, not how they are arranged.

It turns out that the elements of the set need not be related. So, for instance, if I have a set consisting of entities, Narendra Modi, Manmohan Singh, Ashish Choudhury and 100 it is a valid set as far as the definition of a set, because the definition does not say anything regarding the properties of the elements of the same.

Sets can include items that are not inherently related to one another. For example, a set can have both political leaders and a number, such as {Narendra Modi, Manmohan Singh, Ashish Choudhury, 100}. The definition of a set allows for any mix of objects without regard to their types or characteristics.

Imagine a box filled with different items - a toy, a spoon, a book, and a pebble. They belong to different categories, but you can put them all in one box and call it a set of 'random items'. It doesn't matter that the items are unrelated; they can still exist together in a set.

We use some well-known well-defined notations for representing sets. So, we use this notation ∈ for a belongs to A. So, this notation 'belongs to', whenever a is an element of set A we use this notation. And throughout this course, we will follow the notation that we will be using small letters for elements of the sets, and we will be using capital letters for the sets.

In set theory, notation is crucial for clarity and understanding. The symbol '∈' is used to indicate membership; for instance, if 'a ∈ A', it means that 'a' is an element of the set 'A'. Set notation typically uses capital letters to represent sets and small letters for their individual elements. This standardized notation helps in identifying and discussing sets effectively.

Think of a library where books (elements) belong to different sections (sets). We use labels (set notation) to easily identify which books are in which section, making it easier to locate them. Just like how we use letters to denote sections (like 'Fiction' or 'Non-Fiction'), we use specific notations in mathematics for clarity.

Now, how do we express a set? There are two well-known methods. The first method is the Roster method, where we specify the elements of the set within braces. So, for instance if A is a set consisting of 4 elements, then I have listed down the elements of the set A and this is a convenient way of representing a set provided the number of elements in the set is small. If the number of elements in the set is extremely large then it will not be feasible to write down or list down all the elements of the set explicitly. So, that is why we use the second form or second way of expressing a set which is also called as the set builder form and what we do here is that instead of listing down the elements of the set, we write down or state the general property of the elements of the set, which is specifically specified by a predicate function.

There are two primary methods for expressing sets: the Roster method and the Set Builder method. The Roster method lists all elements of a small set explicitly, for example, A = {1, 2, 3}. However, if a set is large or infinite (like all even numbers), using the Roster method becomes impractical. Instead, the Set Builder method describes the properties of the elements within the set, such as A = {x | x is an even number}. This method allows for a clear, concise way to represent sets, especially for infinite sets.

Consider a large collection of fruits. If you wanted to list your fruits using the Roster method, it might take a long time to write each one. Instead, you might use the Set Builder method, saying 'F = {x | x is a fruit}', which conveys that any fruit fits this set without listing them all individually.

We often encounter some special sets. So, a null set or the empty set is one of them and this is the notation ϕ which we use to represent the null set. This is also called as phi set or phi set and it is a set which has no elements. So, you can imagine that a directory which has no files inside it is an example of an empty set. Another special set which we encounter is the singleton set and it is a set which has a single element in it.

In set theory, there are special types of sets that are frequently discussed: the empty set (notated as ϕ) and the singleton set. The empty set is a set that contains no elements at all—imagine a box that has nothing inside it. Conversely, a singleton set contains exactly one element, like a box that holds just one ball. Understanding these sets is fundamental as they form the basis for many operations in set theory.

Picture an empty parking lot; it symbolizes the empty set where no cars are present. Now, envision a parking space with just one car in it; this represents a singleton set. Although one is empty, and the other has one item, both are valid sets in the realm of mathematics.

So, now we will introduce some definitions in the context of sets. So, we start with what we call as equality of sets. So, intuitively if I have two sets A and B they will be called, or they will be considered equal sets if they have the same elements. That means if I have any element present in the set A it is present in B and in the same way any element which is present in B is also present in A there is nothing which is extra present in A or which is extra present in B.

Set equality is a fundamental concept that states two sets, A and B, are equal if they contain exactly the same elements. This definition is intuitive; if every element from set A is found in set B and vice versa, then the sets are considered equal. Furthermore, there is also the concept of subsets: set A is a subset of set B (denoted A ⊆ B) if all elements in A are also in B, meaning no elements in A are unique to it when compared to B.

Imagine two jars filled with candies. If Jar A has candies {red, blue, green} and Jar B also has {red, blue, green}, then the jars are equal. However, if Jar B has one additional candy, like yellow, then Jar A is a subset of Jar B because all the candies in Jar A can be found in Jar B. It's like saying every book in a small shelf can also be found in a larger library.

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