To prove that $B \subset A \cup B$, we need to show that every element of $B$ is also an element of $A \cup B$.

Proof:

1. Let $x \in B$:
   This means that $x$ is an element of the set $B$.

2. Definition of Union:
   The set $A \cup B$ consists of all elements that are either in $A$ or in $B$ (or both). Formally:
   $A \cup B = \{ x \mid x \in A \text{ or } x \in B \}$

3. Since $x \in B$, by the definition of union, we have: $x \in A \cup B$ because if $x \in B$, it satisfies the condition to be in $A \cup B$.

4. Conclusion:
   Since $x \in B$ implies $x \in A \cup B$, it follows that every element of $B$ is also an element of $A \cup B$. Therefore, $B \subset A \cup B$.

Final statement:

Thus, we have shown that $B \subset A \cup B$.

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Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. What is the difference between $A \subset B$ and $A = B$?
2. Can you find a counterexample where $B \subset A$ does not hold?
3. What does $A \cap B$ represent and how is it different from $A \cup B$?
4. How do we prove that $A \cup B \subset A \cup B$ is always true?
5. What are the set identities involving union and intersection?

Tip: Always remember that proving set inclusion involves showing that every element of one set belongs to the other!