Introduction to Sets and Representations

What is a Set?

A set is a well-defined collection of objects. This concept is a fundamental part of modern mathematics and is used in almost every branch, including relations, functions, geometry, and probability. The theory was developed by German mathematician Georg Cantor.

What does "well-defined" mean? It means that we can definitively decide whether a given object belongs to the collection or not.

• Example of a set: The collection of vowels in the English alphabet, namely, a, e, i, o, u.
• Example of what is NOT a set: The collection of the five most renowned mathematicians of the world. The criterion for "most renowned" may vary from person to person, making it not well-defined.

Key Terminology and Notation

• Elements: The objects in a set are also called its elements or members.
• Set Notation: Sets are usually denoted by capital letters, e.g., A, B, C, X, Y.
• Element Notation: Elements are usually denoted by small letters, e.g., a, b, c, x, y.
• Belongs to ($\in$): If 'a' is an element of a set A, we write $a \in A$.
• Does not belong to ($\notin$): If 'b' is not an element of a set A, we write $b \notin A$.

Representations of a Set

There are two primary methods for representing a set:

1. Roster or Tabular Form In this form, all the elements of the set are listed, separated by commas, and enclosed within braces {}.

• Example: The set of prime factors of 210 is $\{2, 3, 5, 7\}$.
• Important Rules:
  • The order in which elements are listed is immaterial.
  • An element is not generally repeated.

2. Set-Builder Form In this form, we write a variable (like x) representing any member of the set followed by a property or rule that all elements of the set satisfy.

• The colon ':' stands for "such that".
• Example: $V = \{x : x \text{ is a vowel in the English alphabet}\}$

Example 1: Equation Solution in Roster Form

Question: Write the solution set of the equation $x^2 + x - 2 = 0$ in roster form.

Solution: The given equation can be written as $(x-1)(x+2) = 0$. The solutions are $x=1$ and $x=-2$. Therefore, the solution set in roster form is $\{1, -2\}$.

Example 2: From Roster to Set-Builder Form

Question: Write the set $A = \{1, 4, 9, 16, 25, ...\}$ in set-builder form.

Solution: We can see that the elements are the squares of natural numbers. So, we can write the set as:

$$A = \{x : x = n^2, \text{ where } n \in \mathbb{N}\}$$