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Understanding the union of sets is crucial in set theory. Imagine you have two collections of objects, let's call them Set A and Set B. The union of these two sets is a new set that includes every object that is in Set A, Set B, or in both. This is symbolized as A ∪ B. An important point to remember is that duplications are not allowed in sets, so each object is unique.

For example, if Set A contains {1, 2, 3} and Set B contains {3, 4, 5}, the union A ∪ B would be {1, 2, 3, 4, 5}, notice how the '3' is not repeated. This is the fundamental idea behind the second step of the solution, where sets M and W were united to form a larger set without duplicate elements.

The concept of set equivalence plays a pivotal role in comparing sets. Two sets are said to be equivalent if they have exactly the same elements, no more and no less. It does not matter if the elements are in a different order because in set theory, the order of elements is not significant.

Let's say Set C = {apple, banana, cherry} and Set D = {cherry, banana, apple}; despite the different arrangement, these sets are equivalent because they contain the same items. This idea is central to the third step of the solution, where the equivalency of the united sets M and W is checked against the elements of set D.

Mathematical sets are foundational to understanding many concepts in mathematics and logic. A set is a collection of distinct objects, considered as an object in its own right. These objects are called 'elements' or 'members' of the set. Sets are typically denoted with capital letters and enclosed in curly brackets. For instance, Set E could be represented as {4, 5, 6}.

There are also several special sets, such as the universal set, which contains all objects under consideration, and the empty set, which contains no objects. Understanding the nature of sets and how they can be manipulated is key to solving problems like the one in the exercise, where identifying and combining elements from sets M and W was the primary task.

In mathematics and logic, deductive reasoning is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. It involves the application of general laws to specific instances. If the initial premises are true and the reasoning is correct, the conclusion drawn must also be true.

For example, consider the premises: All humans are mortal. Socrates is a human. Using deductive reasoning, we can conclude that Socrates is mortal. In the context of the exercise solution, deductive reasoning allows us to conclude that if the united set from M and W has the same elements as set D, then they are equivalent, as dictated by the properties of sets.