Understanding Set Theory Identities

The question asks us to identify which of the given statements involving sets and their operations (union, intersection, complement) is incorrect. We are given that A, B, and C are subsets of a Universal set, and A' denotes the complement of set A.

Analyzing Each Set Statement

Let's examine each option provided and determine its validity using known set theory laws and properties.

1. Statement 1: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

   This statement represents the Distributive Law of Union over Intersection. This is a fundamental and correct identity in set theory.

   Think of it like distributing the $\cup A$ operation over the terms inside the parenthesis $(B \cap C)$.

2. Statement 2: $A' \cup (A \cup B) = (B' \cap B)' \cup A'$

   Let's simplify both sides of this equation.

   Left Side (LHS): $A' \cup (A \cup B)$

   • Using the Associative Law for Union: $(A' \cup A) \cup B$
   • Using the Complement Law ($A' \cup A = U$, where U is the Universal set): $U \cup B$
   • Using the Identity Law ($U \cup X = U$ for any set X): $U$

   So, the LHS simplifies to $U$.

   Right Side (RHS): $(B' \cap B)' \cup A'$

   • Using the Complement Law ($B' \cap B = \emptyset$, where $\emptyset$ is the empty set): $(\emptyset)' \cup A'$
   • Using the Complement Law for the empty set ($(\emptyset)' = U$): $U \cup A'$
   • Using the Identity Law ($U \cup X = U$): $U$

   So, the RHS simplifies to $U$.

   Since LHS = RHS ($U = U$), this statement is a correct identity.

3. Statement 3: $A' \cup (B \cup C) = (C' \cap B)' \cap A'$

   Let's simplify the RHS using De Morgan's Laws and other properties.

   Right Side (RHS): $(C' \cap B)' \cap A'$

   • Using De Morgan's Law ($(X \cap Y)' = X' \cup Y'$): $(C')' \cup B'$
   • Using the Double Complement Law ($(X')' = X$): $C \cup B'$

   Substituting this back into the RHS expression, we get: $(C \cup B') \cap A'$.

   Now the statement is $A' \cup (B \cup C) = (C \cup B') \cap A'$.

   The LHS is a union of $A'$ with $(B \cup C)$, while the simplified RHS is an intersection of $A'$ with $(C \cup B')$. These two forms are generally not equivalent. For example, the LHS contains all elements in $A'$, but the RHS only contains elements that are both in $A'$ AND in $(C \cup B')$. If $B \cup C$ contains elements not in $A'$, the LHS will include them, but the RHS will not. This indicates the statement is likely incorrect.

   To be certain, let's use a simple counterexample.

   Let Universal set $U = \{1, 2, 3, 4\}$, $A = \{1\}$, $B = \{2\}$, $C = \{3\}$.

   • $A' = \{2, 3, 4\}$
   • $B \cup C = \{2\} \cup \{3\} = \{2, 3\}$

   LHS: $A' \cup (B \cup C) = \{2, 3, 4\} \cup \{2, 3\} = \{2, 3, 4\}$.

   • $C' = \{1, 2, 4\}$
   • $C' \cap B = \{1, 2, 4\} \cap \{2\} = \{2\}$
   • $(C' \cap B)' = \{2\}' = \{1, 3, 4\}$

   RHS: $(C' \cap B)' \cap A' = \{1, 3, 4\} \cap \{2, 3, 4\} = \{3, 4\}$.

   Since LHS $\{2, 3, 4\} \neq$ RHS $\{3, 4\}$, the statement $A' \cup (B \cup C) = (C' \cap B)' \cap A'$ is not correct.

4. Statement 4: $(A \cap B) \cup C = (A \cup C) \cap (B \cup C)$

   This statement represents the Distributive Law of Intersection over Union, but with the operations swapped compared to the usual form. It is still a correct identity in set theory.

   Think of it like distributing the $\cup C$ operation over the terms inside the parenthesis $(A \cap B)$.

Based on the analysis, Statement 3 is the one that is not a correct set identity.

Conclusion on Incorrect Statement

After examining all four statements and applying set theory laws and a counterexample, we found that the statement $A' \cup (B \cup C) = (C' \cap B)' \cap A'$ is not a correct identity. The other three statements represent valid set theory laws (Distributive Laws and simplifications leading to Universal set U).

Revision Table: Key Set Theory Laws

Understanding basic set theory laws is crucial for solving problems like this.

Additional Information on Set Identities and Verification

Set identities are equations involving sets that are true for any subsets of a given universal set. They are similar to algebraic identities.

There are several ways to verify if a set identity is correct:

• Using Set Laws: Start from one side of the equation and use known set theory laws (like those listed above) to transform it step-by-step into the other side. This is a rigorous method.
• Using Membership Table: Create a table showing whether an arbitrary element 'x' belongs to each set (A, B, C) and then evaluate whether 'x' belongs to the set represented by the LHS and the RHS of the identity. If the membership is the same for all possible cases, the identity is correct.
• Using Venn Diagrams: Draw Venn diagrams for both sides of the equation and shade the regions represented by the expressions. If the shaded regions are identical, the identity is correct. This method is intuitive but might be less rigorous for complex identities.
• Using Counterexample: To show that a statement is *not* correct, find specific sets for A, B, and C within a universal set such that the LHS and RHS of the equation evaluate to different sets. A single counterexample is sufficient to prove an identity incorrect.

In this problem, we used simplification via laws for statements 1, 2, and 4, confirming they are correct. For statement 3, simplification showed it was likely incorrect, and a counterexample definitively proved its incorrectness.