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Discrete mathematics is a branch of mathematics that deals with objects that can be distinctly separated and counted. It involves studying mathematical structures that are fundamentally discrete rather than continuous. In other words, it deals with countable, separated values rather than smooth and continuous values.

Within discrete mathematics, set theory is a foundational part, and understanding sets is crucial for grasping the broader concepts involved. It applies heavily in computer science, coding theory, and combinatorics, among other areas. As we explore the nature of sets, such as the ones presented in the exercise with elements {Will, Smith}, we venture into one of discrete mathematics' fundamental components.

Sets are collections of distinct objects, known as elements, and determining the equality of sets hinges on the elements they contain, not on the order of these elements. Students studying discrete mathematics will often encounter problems related to set equality, which is a key concept in understanding how sets relate to each other.

The elements of a set are the objects or members that make up that set. In the context of the given exercise, the elements of our sets are 'Will' and 'Smith'.

Elements are the building blocks of sets, and it's important for students to recognize that a set is defined by its elements. In our exercise, the elements are names, but in other scenarios, elements can be numbers, points, other sets, and so on. An essential aspect of elements is that duplication does not matter in a set - if Will is listed twice in a set, it's still considered to have just one 'Will' element.

Moreover, when we list elements or describe sets, such as {Will, Smith} or {Smith, Will}, we can see that although the order in which we list the names may change, the sets remain fundamentally the same because they have precisely the same elements. This is crucial when determining set equality.

Understanding the properties of sets is integral to working with them effectively in various fields of mathematics and science. One of the core properties of sets, demonstrated in the exercise, is that sets are unordered: The set {Will, Smith} is indeed the same as the set {Smith, Will} because sets do not consider the order of elements.

Another important property is that sets are collections of unique elements, meaning repetitions are not counted separately. This ties back to the concept of elements of a set - no matter how many times 'Will' might appear in a set, it only counts once.

Furthermore, when we talk about the equality of sets, we're saying that two sets have exactly the same elements, no more, no less. In terms of set operations, properties such as union, intersection, and difference also become vitally important, but those go beyond the scope of the current discussion on set equality.