Given the expression $(A \cup B) \cap (A \cup B') = A$, let's break it down step-by-step using set theory:

1. Definitions:

• $A \cup B$ represents the union of sets $A$ and $B$.
• $A \cup B'$ represents the union of set $A$ with the complement of $B$ (where $B'$ is the complement of $B$).
• The intersection symbol $\cap$ represents the common elements between the sets.

2. Expression Breakdown:

We need to determine the result of $(A \cup B) \cap (A \cup B')$ and show that it equals $A$.

Step 1: Expand both unions.

• $A \cup B$ includes all elements that are in $A$, in $B$, or in both.
• $A \cup B'$ includes all elements that are in $A$, or not in $B$.

Step 2: Consider the intersection.

• The intersection $(A \cup B) \cap (A \cup B')$ includes all elements that are common to both $A \cup B$ and $A \cup B'$.

Step 3: Analyze.

• Any element in $A$ will be included in both $A \cup B$ and $A \cup B'$, so these elements will definitely be in the intersection.
• For any element $x$ not in $A$:
  • If $x$ is in $B$, it is in $A \cup B$, but not in $A \cup B'$ (since it would not be in $B'$).
  • If $x$ is in $B'$, it is in $A \cup B'$, but not in $A \cup B$.

Thus, no elements outside $A$ can satisfy both $A \cup B$ and $A \cup B'$. Therefore, only the elements of $A$ are included in the intersection.

Conclusion:

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

This shows that the given expression is indeed true.

Would you like more details on this, or do you have any questions?

Related Questions:

1. What is the distributive property of sets, and how does it apply to unions and intersections?
2. How can Venn diagrams be used to visualize the expression $(A \cup B) \cap (A \cup B') = A$?
3. What is the significance of De Morgan's laws in set theory?
4. Can the expression be proven using a Venn diagram or algebraic approach?
5. How does the complement operation affect the union and intersection of sets?

Tip:

When working with set theory expressions, try visualizing the sets using Venn diagrams. This often provides clear insights into relationships between the sets.