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The roster method sometimes referred to as the tabular form, is a straightforward and commonly used way to define a set in mathematics. In this approach, elements are listed explicitly within curly braces, separated by commas, without repetition and without regard to order. For example, the set of natural numbers less than 4 can be represented using the roster method as \( \{1, 2, 3\} \).

This method is particularly useful when dealing with finite sets, or sets with a limited number of elements, as it provides a clear picture of the set's contents. It's important to note that even if we listed the elements as \( \{3, 1, 2\} \), it would represent the same set since the order of elements in set notation is not significant.

While the roster method is a great tool for small or finite sets, it becomes impractical for very large or infinite sets. In such cases, set-builder notation or other methods might be more appropriate.

Set theory is the mathematical science that examines the properties of sets, which are collections of objects that share a common property. These objects could be numbers, characters, or even other sets, and are known as elements or members of the set. When we talk about the 'set of natural numbers less than 4,' we are applying set theory to analyze and understand the collection of numbers that fit this criteria.

In set theory, two sets are considered equal if they contain the same elements, regardless of how they are described or in what order the elements are listed. Moreover, set theory deals with operations involving sets, like unions, intersections, and complements, and provides a foundation for many other areas of mathematics. By studying sets and their relationships, we lay the groundwork for understanding more complex mathematical concepts.

Mathematical sets are the bread and butter of set theory. They are defined as a well-defined collection of distinct objects, where 'well-defined' means that it is clear whether an object belongs to the set or not. For instance, the set of natural numbers less than 4 is well-defined because any natural number can be unambiguously checked to see if it is less than 4 or not. Sets can be infinite, like the set of all natural numbers, or finite, like the set \( \{1, 2, 3\} \).

The elements of a set are usually written between curly braces, and as we have seen, the roster method would list them out, such as \( \{a, b, c\} \) for a set with elements 'a', 'b', and 'c'. When talking about sets, it's also important to consider the concept of subsets and cardinality. A subset is a set every member of which is also a member of another set, and the cardinality of a set refers to the number of elements in it. Understanding these basic terms and principles is essential for grasping higher-level mathematical topics.