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In set theory, the union of two sets is a fundamental operation that combines all elements from each set. For example, if we have two sets, \(A\) and \(B\), the union, denoted as \(A \cup B\), will include every element that belongs to either \(A\) or \(B\). It essentially collects every unique element from both sets into a single set.

In our exercise, set \(A\) is \(\{0, 1, 2, 3, 4\}\), and set \(B\) is \(\{-3, -2, -1\}\). To find their union, we gather all elements from \(A\) and \(B\) without repeating any numbers, since each number in a set must be unique.

As a result, the union of these two sets is \(A \cup B = \{-3, -2, -1, 0, 1, 2, 3, 4\}\). This union includes every individual number from both sets combined into one comprehensive set without duplicates.

Disjoint sets are sets that have no elements in common. In other words, when you look at two sets and find that their intersection is an empty set, they are considered disjoint.

In our example, set \(A\) includes numbers \(0\) through \(4\), while set \(B\) comprises negative integers \(-3\), \(-2\), and \(-1\). Notice that there is no overlap in these numbers. Since \(A\) and \(B\) have no shared elements, they can be described as disjoint.

This property of being disjoint is crucial when verifying certain equations in set theory, such as determining the count of unique elements in the union of the sets. It simplifies verification because the union of disjoint sets \(A\) and \(B\) equals the sum of their individual cardinalities.

Cardinality is the term used to describe the number of elements within a set. It's a simple yet essential aspect of set theory that helps us analyze and understand collections.

Consider set \(A\), which has elements \(\{0, 1, 2, 3, 4\}\), so its cardinality, or the number of elements, is \(n(A) = 5\). Similarly, set \(B\) has elements \(\{-3, -2, -1\}\), giving it a cardinality of \(n(B) = 3\).

When we determine the union \(A \cup B = \{-3, -2, -1, 0, 1, 2, 3, 4\}\), its cardinality turns out to be \(n(A \cup B) = 8\). Understanding cardinality is fundamental when verifying the cardinality of unions of any sets and evaluating equations related to them.

Verification of equations in set theory involves checking the accuracy or correctness of mathematical statements or identities involving sets. It's a way of ensuring that the relationships we assume about sets hold true.

In the problem, we have the equation \(n(A \cup B) = n(A) + n(B)\) to verify. Because sets \(A\) and \(B\) are disjoint, their union's cardinality should indeed equal the sum of their individual cardinalities.

By substituting our earlier findings: \(n(A) = 5\), \(n(B) = 3\), and \(n(A \cup B) = 8\), we can see that \(8 = 5 + 3\) perfectly satisfies the equation. This confirms that our understanding of set operations like the union and properties like cardinality is correctly applied when dealing with disjoint sets. Verification gives us confidence in both our calculations and our theoretical understanding.