Introduction to Sets

Sets were a powerful addition to the arsenal of mathematicians and sets still play a powerful role at the forefront of mathematics. Sets do tend to be more theoretical rather than practical, but Set Theory is an interesting field with important consequences if you want to explore that side.

So, what are sets and why do mathematicians need them?

We need to group similar objects together to make sense of them. Grouping similar objects allows us to treat them in simpler and more powerful manner and gain insights that would have not been possible if we were treating these objects individually.

First, what are sets? You already use them and know some of them. Natural numbers, Integers, Rational Numbers, Transcendental Numbers, Real Numbers, Irrational Numbers (all of which were introduced in Unit - 1) are groups of similar numbers. They are all sets.

Some Use Cases of Sets

Sets are foundational to mathematics and computation. All modern programming languages have inbuilt functions to work with sets because of how good they are at organizing information. Sets play a role in how fast Google is able to return search results in milliseconds. Sets are instrumental in creating databases that serve as the storage for all of internet.

There are advanced mathematical theorems that have been made possible through the power of sets. Sets have greatly expanded our concept of infinity. We have used sets to figure out what can and cannot be proved within mathematics itself using tools like Forcing.

Definition of a Set

Mathematically, a set a well-defined collection of distinct objects. This is it. There are only two conditions.

• Well-defined
• Distinct Objects

Well-defined means there shouldn't be any ambiguity whether one item belongs in a set or not. Distinct means there can only be one of each object in a set, a set cannot have duplicates. The duplicate value in a set is simply ignored.

Examples of a Set

You can create and describe a set yourself in this exercise. Use the dropdowns to build a random set.

This is all sets have to be for now. Any set that you create, whether you leave it empty or put 1, 2 or 3 things inside it, all of them are valid sets. Sets are just a group of distinct well defined objects that you want to put together.

What Can Go Inside a Set?

Any thing can be an element of a set. A set can have numbers, alphabets, words, equations, symbols, and even other sets as its element. $A = { 1, {1} }$ has two elements. It has one number and one set with one number. $B = { 1, {1, 2} }$ also has only two elements, one number and one set with two numbers.

Some Common Sets

$R = Q \cup Q'$ represents the idea that the real number set is the Union or the Rational and Irrational Number set. We will see the operation Union in a later lesson but it is simply combining two sets together to make a new set, the new set has all the elements from both sets.

$Q = \left\{x \mid x = \dfrac{p}{q} \text{ where } p, q \in Z, q \neq 0 \right\}$ this is the set builder notation to describe sets that are difficult to put explicitly write down. This statement is read as, "The set Q has elements x such that x is equal to p over q where p and q belong to the Integer Set and q is not equal to zero."

Three ways to describe a Set

Consider that you create a set named V of all the english vowels, a, e, i, o and u. How would you tell someone what exactly is in your set. There are generally three methods.

• Descriptive Method - "V is a set of all vowels of the English alphabet.
• Tabular Method - $V = \{ a, e, i, o, u\}$
• Set-Builder Method - $V = \{x | x \text{ belongs to the English alphabet and x is a vowel}\}$

The descriptive and tabular method are self explanatory, one involves describing the set and the second involves explicitly writing down all the elements of a set. Set-builder is something new and different but really easy to understand.

Elements of Set-Builder Notation

The set-builder notation is useful when you are creating a smaller set from a larger set. A more complete example of a set-builder notation has the following parts.

• Identify an arbitrary member of a larger set (x is an English alphabet)
• State the property common to the new set (x is a vowel)

Set builder notation is the language of the mathematicians. It is more powerful than descriptive and the tabular method both. The next lesson will be all about creating set-builder notations for different sets.

Activity: Match The Columns!

Reorder the Definitions to match them with the correct Terms.