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Set theory is the mathematical science of sets, which are collections of objects that share a common property. The objects in a set are called elements, and a set can be defined by simply listing its elements between curly brackets. For example, a set of even numbers less than 10 can be represented as \( \{2, 4, 6, 8\} \).

In set theory, various operations can be performed with sets, such as union, intersection, and relative complement. The relative complement or difference between two sets A and B, denoted as \( A \setminus B \), consists of elements that are in A but not in B. It's also important to note that relative complements depend on what the 'universe' consists of, as seen in the textbook example.

Exploring the concept of relative complements further, if we have multiple sets within a universal set, the complement may vary depending on which set it is being compared to. This provides a diverse scope for understanding different relationships within a set structure and offers a practical ground for applying mathematical logic.

The universal set, in set theory, is the set that contains all the objects under consideration for a particular discussion or problem. Usually denoted by U or sometimes E, it encompasses every possible element that can be included or referred to. No set can be broader than the universal set in a given context, and every other set is a subset of this universal set.

For instance, if the universal set \( U \), is defined as \( U = \{1, 2, 3, 4\} \), then any other set composed of these numbers, like \( A = \{1, 2\} \) or \( B = \{2, 3\} \) is considered to be a subset of U. The concept of the universal set is crucial as it provides the relative complement with its boundary. Without the universal set, the idea of 'everything else' that is not in set A would not have a defined meaning, as illustrated in the exercise provided.

Mathematical logic is a subfield of mathematics exploring formal systems in relation to the way we think. It helps to understand and elucidate the principles of reasoning, employing abstract symbols to represent both mathematical statements and the logical structure of mathematics itself.

In the context of set theory, mathematical logic is applied to reason about the properties of sets and the relationships between them. Operations like the relative complement are viewed through the lens of logical formulas, such as \( x \in A \setminus B \) if and only if \( x \in A \) and \( x \otin B \). The textbook example shows that the definition of complementarity relies on underlying logical rules—distinct for each pairing of sets and their respective universal set. Such logical rules are essential to ensure consistency within mathematics and for solving complex problems with clarity and precision.