2. Give a general prove of the statement below.
Let X and Y be two non-empty sets. If Y ⊆ X, then show that X ∪ Y = X.

To prove the statement "If \( Y \subseteq X \), then \( X \cup Y = X \)" for any two non-empty sets \( X \) and \( Y \), we will use the definitions of set union and subset.

### Definitions:
1. **Union of Sets**: The union of two sets \( X \) and \( Y \), denoted \( X \cup Y \), is the set of all elements that are in \( X \), in \( Y \), or in both.
   \[
   X \cup Y = \{ z \mid z \in X \text{ or } z \in Y \}
   \]
2. **Subset**: A set \( Y \) is a subset of a set \( X \), denoted \( Y \subseteq X \), if every element of \( Y \) is also an element of \( X \).
   \[
   Y \subseteq X \iff (\forall y \in Y, y \in X)
   \]

### Proof:

To prove that \( X \cup Y = X \) when \( Y \subseteq X \), we need to show two inclusions:
1. \( X \cup Y \subseteq X \)
2. \( X \subseteq X \cup Y \)

#### 1. Prove \( X \cup Y \subseteq X \):

Let \( z \) be an arbitrary element of \( X \cup Y \). By the definition of union:
\[
z \in X \cup Y \implies z \in X \text{ or } z \in Y
\]
We need to show that \( z \in X \).

- If \( z \in X \), then \( z \in X \).
- If \( z \in Y \), since \( Y \subseteq X \), \( z \in X \).

In both cases, we have \( z \in X \). Therefore,
\[
z \in X \cup Y \implies z \in X
\]
Hence, \( X \cup Y \subseteq X \).

#### 2. Prove \( X \subseteq X \cup Y \):

Let \( z \) be an arbitrary element of \( X \). By the definition of union:
\[
z \in X \implies z \in X \cup Y
\]
because \( z \in X \) satisfies the condition for being in the union \( X \cup Y \).

Thus,
\[
z \in X \implies z \in X \cup Y
\]
Hence, \( X \subseteq X \cup Y \).

### Conclusion:
Since we have shown both \( X \cup Y \subseteq X \) and \( X \subseteq X \cup Y \), it follows that:
\[
X \cup Y = X
\]

This completes the proof.

---

Would you like more details on any part of this proof or have any questions?

Here are some related questions you might find interesting:
1. How do you prove that \( X \cap Y = Y \) if \( Y \subseteq X \)?
2. What is an example where \( Y \not\subseteq X \) and \( X \cup Y \neq X \)?
3. How do you define the intersection of two sets?
4. Can the empty set be a subset of any set? Why?
5. How would you prove that the union of a set with the empty set is the set itself?
6. What is the symmetric difference of two sets and how is it defined?
7. How do you prove that \( X \cup (X \cap Y) = X \)?
8. What are the properties of set union and intersection that are used in proofs?

**Tip**: When working with sets, always start by carefully defining the sets and the operations you are using. This helps in constructing clear and rigorous proofs.

Learn how to prove the statement "If Y is a subset of X, then X union Y equals X" using set theory. This proof involves understanding set union and subset definitions, and demonstrates the equality in set operations. Suitable for advanced students and learners of set theory.

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