We are tasked with proving the following two set identities:

1. $A \cup B = B \cup A$
2. $A \cap B = B \cap A$

1. Proving $A \cup B = B \cup A$

To prove that $A \cup B = B \cup A$, we need to show that every element in $A \cup B$ is also in $B \cup A$, and vice versa.

Step 1: Prove $A \cup B \subseteq B \cup A$

• Let $x \in A \cup B$. By definition of the union of sets, this means $x \in A$ or $x \in B$.
  • If $x \in A$, then $x \in B \cup A$ (since $x$ is in $A$, and $B \cup A$ includes all elements of $A$).
  • If $x \in B$, then $x \in B \cup A$ (since $x$ is in $B$, and $B \cup A$ includes all elements of $B$).
• Therefore, $x \in B \cup A$, proving that $A \cup B \subseteq B \cup A$.

Step 2: Prove $B \cup A \subseteq A \cup B$

• Let $x \in B \cup A$. By definition of the union of sets, this means $x \in B$ or $x \in A$.
  • If $x \in B$, then $x \in A \cup B$ (since $x$ is in $B$, and $A \cup B$ includes all elements of $B$).
  • If $x \in A$, then $x \in A \cup B$ (since $x$ is in $A$, and $A \cup B$ includes all elements of $A$).
• Therefore, $x \in A \cup B$, proving that $B \cup A \subseteq A \cup B$.

Since both inclusions hold, we have shown that $A \cup B = B \cup A$.

2. Proving $A \cap B = B \cap A$

Next, we need to prove that $A \cap B = B \cap A$. Again, we show that every element of $A \cap B$ is in $B \cap A$, and vice versa.

Step 1: Prove $A \cap B \subseteq B \cap A$

• Let $x \in A \cap B$. By the definition of intersection, this means that $x \in A$ and $x \in B$.
• Since $x \in A$ and $x \in B$, it follows that $x \in B \cap A$ (because $B \cap A$ includes all elements that are in both $B$ and $A$).
• Therefore, $x \in B \cap A$, proving that $A \cap B \subseteq B \cap A$.

Step 2: Prove $B \cap A \subseteq A \cap B$

• Let $x \in B \cap A$. By the definition of intersection, this means that $x \in B$ and $x \in A$.
• Since $x \in B$ and $x \in A$, it follows that $x \in A \cap B$ (because $A \cap B$ includes all elements that are in both $A$ and $B$).
• Therefore, $x \in A \cap B$, proving that $B \cap A \subseteq A \cap B$.

Since both inclusions hold, we have shown that $A \cap B = B \cap A$.

Conclusion

We have proven both identities:

1. $A \cup B = B \cup A$
2. $A \cap B = B \cap A$

Would you like more details or further clarification on any part of the proof?

Here are 5 related questions:

1. What are the properties of set unions and intersections in general?
2. How can De Morgan's Laws be applied to set operations?
3. What is the difference between the symmetric difference and union/intersection of sets?
4. How can Venn diagrams help visualize set identities?
5. How do set operations behave in the context of finite and infinite sets?

Tip: When proving set identities, it's helpful to break the problem into smaller inclusions (showing both subsets) to demonstrate equality.