Question

Let 'U' be universal set. If A and B are subsets of U, then match the following and choose the correct answer.

• A. a - ii, b - i, c - iv, d - iii
• B. a - ii, b - iii, c - iv, d - i
• C. a - iii, b - iv, c - ii, d - i
• D. a - iv, b - iii, c - ii, d - i

Hint:

De Morgan’s Laws are easiest to remember as: “Break the line, change the sign” - Union becomes intersection - Intersection becomes union while keeping complements on each term.

Answer 1

The Correct Option is D

Concept: This problem is based on fundamental laws of set theory, mainly the Complement Law, Double Complement Law, and De Morgan’s Laws. These laws help simplify expressions involving union, intersection, and complement of sets.
• Complement Law: For any set \(A\), the union of a set with its complement gives the universal set: \[ A \cup A' = U \]
• Double Complement Law: \[ (A')' = A \]
• De Morgan’s Laws: \[ (A \cup B)' = A' \cap B' \] \[ (A \cap B)' = A' \cup B' \]

Step 1: Evaluate (a) \(A \cup A'\) Every element in the universal set either belongs to \(A\) or does not belong to \(A\). Hence their union covers the entire universal set: \[ A \cup A' = U \] So, (a) matches with (iv).

Step 2: Evaluate (b) \((A \cup B)'\) Using De Morgan’s law, the complement of union becomes intersection of complements: \[ (A \cup B)' = A' \cap B' \] So, (b) matches with (ii).

Step 3: Evaluate (c) \((A \cap B)'\) Using De Morgan’s law again, the complement of intersection becomes union of complements: \[ (A \cap B)' = A' \cup B' \] So, (c) matches with (iii).

Step 4: Evaluate (d) \((A')'\) Applying double complement law: \[ (A')' = A \] So, (d) matches with (i). Final Matching: \[ a \rightarrow (iv), \quad b \rightarrow (ii), \quad c \rightarrow (iii), \quad d \rightarrow (i) \] Thus, the correct option is: \[ \boxed{(D)} \]