The problem in the image asks to prove by double inclusion that:

$A \cap (A \cup B) = A$

Double Inclusion Proof:

To prove that $A \cap (A \cup B) = A$, we need to show two inclusions:

1. $A \cap (A \cup B) \subseteq A$
2. $A \subseteq A \cap (A \cup B)$

1. Prove $A \cap (A \cup B) \subseteq A$

Let $x \in A \cap (A \cup B)$.

• By the definition of intersection, $x \in A$ and $x \in A \cup B$.
• Since $x \in A \cup B$, it means that $x \in A$ or $x \in B$.
• But we already know $x \in A$, so this is sufficient.

Thus, $x \in A$. Therefore, $A \cap (A \cup B) \subseteq A$.

2. Prove $A \subseteq A \cap (A \cup B)$

Let $x \in A$.

• By the definition of union, $x \in A \cup B$ (since $x \in A$).
• Also, since $x \in A$, it satisfies the condition for intersection.

Thus, $x \in A \cap (A \cup B)$. Therefore, $A \subseteq A \cap (A \cup B)$.

Conclusion:

Since both inclusions hold, we conclude that:

$A \cap (A \cup B) = A$

This completes the proof by double inclusion.

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the difference between union and intersection of sets?
2. Can this proof be generalized for more than two sets?
3. How does the distributive property apply to set operations like intersection and union?
4. What other methods (besides double inclusion) can be used to prove set identities?
5. How can Venn diagrams be used to visualize this proof?

Tip: In set theory, the intersection represents elements common to both sets, while the union combines elements from both sets.