Sets - Sets and their Representations

A set is a well-defined collection of objects, meaning there is no ambiguity about whether an object belongs to the collection or not. Imagine a set as a closed loop or a 'container' where every item inside is distinct and clearly identified.

Elements of a set are denoted by lowercase letters ($a, b, c, \dots$) while the set itself is named using uppercase letters ($A, B, C, \dots$). The symbol $\in$ is used to denote 'belongs to', and $\notin$ denotes 'does not belong to'. Visually, if an element $x$ is inside the boundary of set $A$, we write $x \in A$.

Roster or Tabular Form: In this representation, all elements of the set are listed, separated by commas and enclosed within curly braces $\{ \}$. For example, the set of vowels in English is $V = \{a, e, i, o, u\}$. The order of elements does not matter, and repeating an element has no effect on the set.

Set-builder Form: Instead of listing elements, we describe the common property shared by all elements. It is written as $A = \{x : P(x)\}$, which reads as 'the set of all $x$ such that $x$ satisfies property $P$'. Visualize this as a 'filter' or 'rule' that selects specific numbers or objects to be included in the set.

Standard Sets of Numbers: Specific symbols are used for common number systems. $\mathbb{N}$ represents Natural numbers, $\mathbb{Z}$ represents Integers, $\mathbb{Q}$ represents Rational numbers, and $\mathbb{R}$ represents Real numbers. Visualize $\mathbb{N}$ as discrete points on a number line starting from 1, while $\mathbb{R}$ is the entire continuous line.

The Empty Set: A set which does not contain any element is called the empty set or the null set. It is denoted by the symbol $\phi$ or empty braces $\{ \}$. Visually, this is represented by a circle or loop that contains nothing inside it.