UPSC NDA 2 2021 Mathematics Solved Paper

Concept:Distributive law and De Morgan's law are fundamental set operations that help simplify expressions. A set difference being empty only implies subset, not equality.Explanation:Statement 1: Union distributes over intersection. That is, $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$. So statement 1 is correct.Statement 2: De Morgan’s law states that $(A \cup B)' = A' \cap B'$. Hence statement 2 is correct.Statement 3: If the difference $A - B = \phi$, then every element of $A$ is also in $B$ (i.e., $A \subseteq B$). However, $B - A$ may not be empty, so $A$ and $B$ need not be equal. Example: $A = \{2,3\}$, $B = \{2,3,4\}$ gives $A-B=\phi$ but $A \neq B$. Thus statement 3 is incorrect.Answer:Only statements 1 and 2 are correct. Hence the correct option is A.