The given equation is:

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

This statement is true. Let's break it down using set theory properties.

Step-by-Step Explanation:

Left-hand side: $(A \cap B) \cup C$

1. $A \cap B$ represents the elements common to both $A$ and $B$.
2. $(A \cap B) \cup C$ means we take the union of the set $A \cap B$ with $C$, meaning we include all elements in $C$ as well as the elements that are common to both $A$ and $B$.

Right-hand side: $(A \cup C) \cap (B \cup C)$

1. $A \cup C$ represents all elements that are either in $A$ or in $C$.
2. $B \cup C$ represents all elements that are either in $B$ or in $C$.
3. $(A \cup C) \cap (B \cup C)$ represents the intersection of the two unions. This means we only include the elements that are present in both $A \cup C$ and $B \cup C$, which will be the common elements between them.

Why is it true?

These two sides are equivalent because both of them lead to the set containing:

• Elements that are either in $C$, or
• Elements that are in both $A$ and $B$.

In other words, both expressions describe a set that includes:

• All elements of $C$, and
• Any element that is common to both $A$ and $B$.

Therefore, the two expressions are equivalent, making the statement true.

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Would you like more details on any part of this explanation or another related concept?

Here are 5 related questions to explore further:

1. How do the union and intersection operations interact in set theory?
2. Can De Morgan's laws be applied to this equation?
3. How can we use Venn diagrams to visually confirm the equivalence of these two sides?
4. What is the difference between symmetric difference and intersection in set theory?
5. How would the result change if we were dealing with three sets instead of two?

Tip: When unsure about equivalences in set theory, using Venn diagrams is a helpful way to visualize set operations and verify their truth.