Sets - Introduction to Sets: Definition, Roster Form, Set-Builder Form

A set is a well-defined collection of distinct objects. 'Well-defined' means that given a specific object, we can clearly determine whether it belongs to the collection or not. For example, 'the collection of all vowels in the English alphabet' is a set, whereas 'the collection of good cricketers' is not a set because the criteria for 'good' is subjective.

The individual objects within a set are called its members or elements. We use the symbol $\in$ to denote 'is an element of' or 'belongs to', and $\notin$ to denote 'is not an element of'. For instance, if $V = \{a, e, i, o, u\}$, then $e \in V$ but $b \notin V$.

Sets are typically represented by enclosing their elements within curly brackets $\{ \}$ and separating them with commas. Visually, a set looks like a container of items, such as $A = \{1, 2, 3\}$, where the brackets act as the boundary of the collection.

In Roster Form (or Tabular Form), all the elements of the set are listed explicitly. The order of listing elements does not matter, so $\{1, 2, 3\}$ is the same as $\{3, 1, 2\}$. Also, elements are not usually repeated; for example, the set of letters in the word 'APPLE' is written as $\{A, P, L, E\}$.

In Set-Builder Form (or Rule Method), instead of listing elements, we state a property or rule that all elements must satisfy. It is written as $A = \{x : x \text{ satisfies property } P\}$. The colon symbol ':' or a vertical bar '|' is read as 'such that'. For example, $\{x : x \text{ is a natural number and } x < 5\}$ defines the set $\{1, 2, 3, 4\}$.

The Cardinal Number of a finite set $A$ is the number of distinct elements present in it. It is denoted by $n(A)$. For example, if $A = \{2, 4, 6, 8\}$, then the cardinal number $n(A) = 4$ because there are four distinct elements.