Here’s a detailed breakdown of the operations you performed, with explanations:

1. Complement of A (A’):

   • The complement of set $A$, denoted as $A'$, is found by subtracting the elements of $A$ from the universal set $U$.
   • In this case, the universal set $U$ contains all the possible items (e.g., Area Rugs, Accent Rugs, Door Mats, etc.), while set $A$ contains a particular subset of these items.
   • The complement of $A$ is therefore the collection of all items in $U$ that are not in $A$.
   • Result for $A'$: {Area Rugs, Accent Rugs, Door Mats, Mirrors, Shelves, Sconces, Wall Clocks, Wall Art, Accent Pillows, Blankets/Throws, Sheets Sets, Comforters, Kitchen Utensils, Canisters, Kitchen Towels, Vases, Baskets, Plants, Flowers, Books, Table runners, Sculptures, Candleholders, Photo Frames}

2. Set Operation: A ∪ (B ∩ C):

   • Here, you’re combining the union of set $A$ with the intersection of sets $B$ and $C$.
   • Intersection means finding the common elements between sets $B$ and $C$. In this case, since $B$ and $C$ (Home Accents and Wall Décor) have no common elements, $B ∩ C = \emptyset$ (the empty set).
   • The union of set $A$ with the empty set is simply $A$, since adding an empty set doesn’t change the content of $A$.
   • Result for $A ∪ (B ∩ C)$: This operation results in just $A$, i.e., the set of items in $A$.

3. Intersecting A and C:

   • When you intersect sets $A$ and $C$ (let’s assume set $C$ represents, for example, Wall Décor), you find any common elements between $A$ and $C$.
   • According to your example, there are no common elements between $A$ and $C$, so their intersection is empty.
   • Result for $A ∩ C$: $\emptyset$ (empty set)

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Key Concepts in Set Theory from This Example:

1. Complement: The complement of a set contains all elements from the universal set that are not in the given set.
2. Union ( ∪ ): Combines elements from two sets without duplication.
3. Intersection ( ∩ ): Contains only the elements that are in both sets.
4. Empty Set ( ∅ ): The set with no elements, often the result of operations like intersections when there are no common elements.

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Would you like more detailed examples of set operations or any clarifications on other concepts related to this topic?

Next Questions to Explore:

1. What is the difference between a union and a Cartesian product in set theory?
2. How does the distributive property of sets work (e.g., $A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)$)?
3. What happens if we intersect a set with its complement? What is the result?
4. Can we perform set operations on infinite sets? What happens in that case?
5. How does the power set of a set relate to its complement?

Tip:

When solving set operations, visualize or list the elements to help clarify the relationships between sets—this can make complex operations like intersections and complements easier to understand.