Understanding Set Operations and Complements

The question asks us to find the complement of the set expression &[(A − B) ∪(B − C) ∪ (C − A)]', given that U is the universal set and A ∪ B ∪ C = U. Let's break down the expression step by step using fundamental set theory concepts.

Analyzing the Components

The expression involves set differences and unions:

• A − B represents the set of elements that are in set A but not in set B. In set notation, this is A ∩ B'.
• B − C represents the set of elements that are in set B but not in set C. In set notation, this is B ∩ C'.
• C − A represents the set of elements that are in set C but not in set A. In set notation, this is C ∩ A'.

The expression inside the complement is the union of these three sets: (A − B) ∪(B − C) ∪ (C − A).

Let's think about what this union represents using a Venn diagram concept. Consider the three sets A, B, and C within the universal set U. The set (A − B) ∪(B − C) ∪ (C − A) includes elements that are:

• In A but not B (could be in C or not in C)
• In B but not C (could be in A or not in A)
• In C but not A (could be in B or not in B)

Specifically, this union covers the regions:

• Elements in A only ($A \cap B' \cap C'$)
• Elements in B only ($B \cap C' \cap A'$)
• Elements in C only ($C \cap A' \cap B'$)
• Elements in A and B but not C ($A \cap B \cap C'$)
• Elements in B and C but not A ($B \cap C \cap A'$)
• Elements in C and A but not B ($C \cap A \cap B'$)

So, the set (A − B) ∪(B − C) ∪ (C − A) consists of all elements that belong to exactly one or exactly two of the sets A, B, or C.

Considering the Universal Set Condition

We are given that A ∪ B ∪ C = U. This is a crucial condition. It means that every element in the universal set U belongs to at least one of the sets A, B, or C. There are no elements in U that are outside A ∪ B ∪ C.

The set A ∪ B ∪ C can be partitioned into the following disjoint regions:

• A ∩ B' ∩ C' (A only)
• B ∩ C' ∩ A' (B only)
• C ∩ A' ∩ B' (C only)
• A ∩ B ∩ C' (A and B only)
• B ∩ C ∩ A' (B and C only)
• C ∩ A ∩ B' (C and A only)
• A ∩ B ∩ C (A and B and C)

Since U = A ∪ B ∪ C, the universal set U is composed entirely of these seven regions.

The set (A − B) ∪(B − C) ∪ (C − A) includes the first six of these regions (elements belonging to exactly one or exactly two sets). The only region within U = A ∪ B ∪ C that is *not* included in (A − B) ∪(B − C) ∪ (C − A) is the last region: A ∩ B ∩ C, which represents elements belonging to all three sets.

Finding the Complement

The question asks for the complement of [(A − B) ∪(B − C) ∪ (C − A)]. Since U = A ∪ B ∪ C, the complement is the set of elements in U that are *not* in (A − B) ∪(B − C) ∪ (C − A).

Based on our analysis, the only part of U that is not covered by (A − B) ∪(B − C) ∪ (C − A) is the intersection of all three sets, A ∩ B ∩ C.

Therefore, [(A − B) ∪(B − C) ∪ (C − A)]' is equal to A ∩ B ∩ C.

The expression simplifies to the intersection of sets A, B, and C.

Summary of the Set Simplification

Given U is the universal set and A ∪ B ∪ C = U, the expression to simplify is $S' = [(A \setminus B) \cup (B \setminus C) \cup (C \setminus A)]'$.

The set $S = (A \setminus B) \cup (B \setminus C) \cup (C \setminus A)$ represents elements belonging to exactly one or exactly two of the sets A, B, C.

Since $U = A \cup B \cup C$, any element in U must belong to at least one of A, B, or C. The regions of $A \cup B \cup C$ are disjoint parts representing elements in A only, B only, C only, A and B only, B and C only, C and A only, and A and B and C.

The set $S$ covers all these regions except the region where elements are in all three sets, which is $A \cap B \cap C$.

Thus, $S$ and $A \cap B \cap C$ are disjoint sets whose union is $U$.

So, the complement of $S$ in U, denoted by $S'$, is exactly the set $A \cap B \cap C$.

$[(A \setminus B) \cup (B \setminus C) \cup (C \setminus A)]' = A \cap B \cap C$.

The final answer is the intersection of A, B, and C.

Revision Table: Key Set Operations

Additional Information: De Morgan's Laws

De Morgan's laws are useful when dealing with complements of unions and intersections. They state:

• $(A \cup B)' = A' \cap B'$ (The complement of the union is the intersection of the complements).
• $(A \cap B)' = A' \cup B'$ (The complement of the intersection is the union of the complements).

These laws can be extended to more than two sets. For three sets A, B, and C:

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

While we could attempt to use De Morgan's laws directly on the complex expression $[(A \cap B') \cup (B \cap C') \cup (C \cap A')]'$, the geometric/Venn diagram approach based on the condition A ∪ B ∪ C = U provides a more intuitive and straightforward path to the solution in this specific problem.

Understanding the different regions created by intersecting sets is key to solving problems involving complements and set operations within a universal set.