Sets - The Empty Set, Finite and Infinite Sets, Equal Sets

The Empty Set: A set which does not contain any element is called the empty set or the null set or the void set. It is denoted by the symbol $\phi$ or $\{ \}$. Visually, an empty set is represented in a Venn diagram as a region or circle containing no points or markings. It is important to note that $\{0\}$ is NOT an empty set, as it contains the element zero.

Finite Sets: A set which is empty or consists of a definite (countable) number of elements is called a finite set. For example, $W = \{ \text{days of the week} \}$ is a finite set because $n(W) = 7$. Visually, a finite set is shown as a closed boundary containing a specific, fixed number of dots representing its members.

Infinite Sets: A set whose elements cannot be listed by a natural number $n$ for any $n$ is called an infinite set. The counting process for such a set never ends. Examples include the set of natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$. In visual representations, infinite sets are often depicted with an ellipsis ($...$) to show that the pattern of elements extends indefinitely beyond the visible area.

Equal Sets: Two sets $A$ and $B$ are said to be equal if they have exactly the same elements, and we write $A = B$. Otherwise, the sets are said to be unequal ($A \neq B$). Visually, if two sets are equal, their Venn diagram circles would perfectly overlap. Note that the order in which elements are listed or the repetition of elements does not change the set; for example, $\{1, 2, 3\} = \{3, 2, 1, 1, 2\}$.

Cardinal Number of a Finite Set: The number of distinct elements in a finite set $A$ is called its cardinal number and is denoted by $n(A)$. For example, if $A = \{a, e, i, o, u\}$, then $n(A) = 5$. This can be visualized as the total count of distinct items contained within the set's boundary.

Equivalent Sets: Two finite sets $A$ and $B$ are called equivalent if their cardinal numbers are the same, i.e., $n(A) = n(B)$. While equal sets must have the same elements, equivalent sets only need to have the same count of elements. Visually, two different circles in a Venn diagram are equivalent if they contain the same number of dots, even if those dots represent different values.

Representation of Infinite Sets: All infinite sets cannot be described in the roster form. For example, the set of real numbers $\mathbb{R}$ cannot be described in roster form because the elements of this set do not follow any particular pattern that can be shown with an ellipsis. Visually, such sets are often represented as continuous lines or shaded regions on a coordinate plane rather than discrete points.