UPSC NDA 2 2022 Mathematics Solved Paper

Concept:Set equality means two sets contain exactly the same elements.The operations of intersection ($\cap$) and union ($\cup$) do not uniquely determine the second set from the first.A single counterexample is enough to disprove a general statement about sets.Explanation:Evaluate Statement 1: $A \cap B = A \cap C \Rightarrow B = C$.Let $A = \{1, 2, 3, 4, 5\}$, $B = \{1, 2, p, q\}$, and $C = \{1, 2, m, n\}$.Here, $A \cap B = \{1, 2\}$ and $A \cap C = \{1, 2\}$.So, $A \cap B = A \cap C$ is satisfied, but $B \neq C$.Therefore, Statement 1 is false.Evaluate Statement 2: $A \cup B = A \cup C \Rightarrow B = C$.Let $A = \{1, 2, 3, 4\}$, $B = \{1, 2, p, q\}$, and $C = \{p, q\}$.Here, $A \cup B = \{1, 2, 3, 4, p, q\}$ and $A \cup C = \{1, 2, 3, 4, p, q\}$.So, $A \cup B = A \cup C$ is satisfied, but $B \neq C$.Therefore, Statement 2 is false.Since both given statements are disproved by counterexamples, neither is always correct.Answer:Option D (Neither 1 nor 2).