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Understanding the concept of a subset is fundamental in set theory. A set, let's call it Set A, is considered a subset of another set, say Set B, if every single element that is in Set A is also found in Set B. We denote this relationship as \( A \subseteq B \). It's important to note that every set is a subset of itself, since all of its elements are contained within the set. This self-inclusivity is often one of the first properties introduced when learning about subsets.

When you look around at everyday examples, think of a set of fruit being a subset of a larger set containing all types of food. Or, every book on a specific topic within a library is part of the larger set of all books in the library. Just like that, when we say that \( X \subseteq X \cup Y \), we're affirming the intuitive fact that all members of set \( X \) will certainly be found in the combination, or union, of set \( X \) with any other set, including set \( Y \).

The union of two sets merges all the elements from both sets into a new set that contains all unique elements without any duplicates. Mathematically, we denote the union of sets X and Y as \( X \cup Y \). This set includes all the elements that are in X, all the elements that are in Y, and eliminates any repeated elements, since sets are inherently collections of distinct objects.

To visualize this, imagine having two buckets of toys; one is red toys (Set X) and the other is blue toys (Set Y). When we combine these into a big box representing their union (\( X \cup Y \)), we gather all red and blue toys together. If there are toys that happen to be both red and blue, they still enter the box only once. In this way, the union encompasses the complete variety of toys from both buckets.

The method of proof by definition is a direct way of showing that a statement is true by simply showing that it meets the criteria laid out by a definition. In set theory, when you're asked to prove a statement like \( X \subseteq X \cup Y \), all that's needed is to go back to the definitions of 'subset' and 'union of sets.'

By demonstrating that every individual element of Set X is indeed part of the set created by \( X \cup Y \), you have used the definitions to validate the original statement. Effectively, you've shown that there is not even a single element that can be contested and thus have completed what is known as a 'proof by definition.' In class or examinations, this approach is valued because it's both methodical and conclusive, leaning on established concepts to back up claims.