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Set theory is the mathematical science of sets, which are collections of objects or elements that can be clearly defined. These objects are called 'members' or 'elements' of the set. The study of sets and their properties is the foundational language for all of mathematics, influencing fields like logic, statistics, and computational science.

In set theory, particular attention is paid to the nature of the elements and the relations between different sets. For instance, it is crucial to understand the distinctions between subsets, proper subsets, equivalent sets, and equal sets. In the context of our example exercise, we know that set B (business majors) is a subset of set A (all students at the college). This is because every element of set B is also contained within set A, but not necessarily the other way around.

Moreover, set theory explores the size or cardinality of sets, which does not concern the specific values of its elements but rather how many elements exist in the collection. This concept leads us into a deeper discussion on the equivalence and equality of sets.

A mathematical set is a well-defined collection of distinct objects considered as a whole. These objects can be anything from numbers, letters, symbols, to even other sets. There are some key terms and concepts in set theory that are of great importance:

• Elements: The individual objects within a set are called elements.
• Cardinality: This refers to the number of elements in a set, symbolized by the 'absolute value' notation bars, like this: |A|.
• Subset: A set that contains only elements found in another set.
• Power Set: The set of all possible subsets, including the empty set and the set itself.
• Venn Diagram: A visual representation of how sets interact with each other.

In our exercise, set A includes every student in the college, while set B refers explicitly to one group within that broader category. This distinction between set A and set B allows us to visualize the mathematical concept of subsets and emphasize their differences in terms of content and cardinality.

Element cardinality in set theory indicates the number of elements in a set. It is a measure of a set's size, irrespective of what the elements are. Cardinality is denoted as |A| for a set A, and it represents a way to compare the size of sets without needing to know the exact elements within them.

For example, we might say two sets are equivalent if their cardinalities are the same, |A| = |B|. However, equivalent sets need not have identical elements, just an equal number of them. To determine if sets A and B from our example are equivalent, we would compare their cardinalities. Since set A includes all students and set B includes only business students, we can deduce that |A| > |B|. Consequently, these sets are not equivalent.

Understanding element cardinality helps us better conceptualize notions such as infinite sets, countable, and uncountable sets in advanced mathematical topics. Grasping this concept is instrumental in solving problems related to set sizes and realizing that 'bigger' or 'smaller' does not refer to physical size but to the quantity of distinct elements within the sets.