We are tasked with proving that if $A \subseteq B$, then $A \cup B = B$. Let's break it down step by step.

Definitions:

• $A \subseteq B$ means that every element of $A$ is also an element of $B$.
• $A \cup B$ denotes the union of sets $A$ and $B$, which contains all elements that are in $A$, $B$, or both.

Proof:

We need to show that $A \cup B = B$ under the assumption $A \subseteq B$.

Step 1: Show that $A \cup B \subseteq B$

• By the definition of union, $A \cup B$ contains elements that are in $A$, $B$, or both.
• But we know $A \subseteq B$, meaning every element of $A$ is already in $B$.
• Thus, all elements of $A \cup B$ are in $B$, since any element in $A$ is already in $B$, and any element in $B$ is obviously in $B$.
• Therefore, $A \cup B \subseteq B$.

Step 2: Show that $B \subseteq A \cup B$

• By the definition of union, $A \cup B$ contains all elements of $B$, because $A \cup B$ includes any element that is in $B$.
• Therefore, every element of $B$ is in $A \cup B$.
• Thus, $B \subseteq A \cup B$.

Step 3: Conclude that $A \cup B = B$

• From Step 1, we have $A \cup B \subseteq B$.
• From Step 2, we have $B \subseteq A \cup B$.
• Since both inclusions hold, we conclude that $A \cup B = B$.

Final Conclusion:

If $A \subseteq B$, then $A \cup B = B$.

Would you like more details or have any questions?

Here are 5 questions to consider:

1. What is the difference between $A \subseteq B$ and $A \subset B$?
2. Can you think of an example where $A \cup B = B$, even if $A \neq B$?
3. How does the union operation behave with disjoint sets?
4. What happens to the union when $A \cap B = \emptyset$?
5. Can $A \cup B = A$ ever be true if $A \subseteq B$?

Tip: When proving set equalities, it is often useful to show mutual inclusion — that each set is a subset of the other.