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In the realm of mathematics, different branches focus on varying aspects of the scientific study of numbers, quantities, and space. Among these, we find finite mathematics, a field which encompasses several disciplines but circumspectly focuses on areas that do not require the concept of continuity. That's to say, unlike calculus, which deals extensively with continuous variables, finite mathematics concentrates on topics such as probability, statistics, finance, and matrics amongst others—all of which can be handled in a discrete manner.

For students delving into problems of finite mathematics, it’s crucial to grasp that this field often deals with finite sets, structures that are impeccably suited for solving real-world problems where the concept of infinity is not practical. For example, when counting the possible outcomes of a dice roll or analyzing voter preferences in an election, you’re working within the bounds of finite mathematics. By considering and analyzing finite scenarios, insights can be drawn without the complications introduced by infinite series or integrable functions.

Consider set theory as the cornerstone of modern mathematics. It's the study of sets, which are essentially collections of objects. These objects could be numbers, people, or even other sets, and they are called the elements or members of a set. When we define a set, for example, using curly braces like in our exercise (\(S = \big{1,2,3,4,5,6\big}\)), we're establishing the confines of that specific collection.

One simple yet fundamental concept of set theory is that it allows us to talk about groups of items or events in a mathematically rigorous way, paving the way for the application of more complex operations like unions, intersections, and complements. These set-based operations reflect how we can combine, compare, and differentiate between groups in real and abstract contexts. Much of set theory feeds directly into the study of logic, probability, and algebra, where understanding how sets interact is foundational.

The intersection of sets is analogous to finding a common ground. When two sets intersect, it’s equivalent to asking, 'What do these sets share?' Mathematically, the intersection is represented by the symbol \( \cap \). It denotes the set of all elements that belong to both sets being intersected.

Let's take our previous example of sets \(F = \big{1,3,5\big}\) and \(G = \big{5,6\big}\). To identify the intersection \(F \cap G\), we seek the elements they have in common. In this case, the element 5 appears in both sets, thus \(F \cap G = \big{5\big}\). This concept is a foundational part of set theory and immensely important in realms like logic, where intersection helps to reveal shared properties or characteristics across different entities or datasets.