Q162 / 270 · NDA 2015 (Phase 2) · Mathematics

$\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D are four sets such that $\mathrm{A} \cap \mathrm{B}=\mathrm{C} \cap \mathrm{D}=\phi$. Consider the following : 1. $\mathrm{A} \cup \mathrm{C}$ and $\mathrm{B} \cup \mathrm{D}$ are always disjoint. 2. $\mathrm{A} \cap \mathrm{C}$ and $\mathrm{B} \cap \mathrm{D}$ are always disjoint Which of the above statements is/are correct?

  • A
    1 only
  • B
    2 only
  • C
    Both 1 and 2
  • Correct answer
    D
    Neither 1 nor 2

Solution

Let $A=\{1,2\}$ $\begin{aligned} & \mathrm{B}=\{3,4,0\} \\ & \mathrm{C}=\{5,6,0\} \\ & \mathrm{D}=\{7,8\} \end{aligned}$ Such that $(A \cap B)=(C \cap D)=\phi$ $\begin{aligned} & \Rightarrow(A \cup C)=\{1,2,5,6,0\} \\ & \Rightarrow(B \cup D)=\{3,4,7,8,0\} \\ & \Rightarrow(A \cup C) \cap(B \cup D)=\{0\} \end{aligned}$ So $(A \cup C)$ and $(B \cup D)$ are not always dispoint $\Rightarrow(\mathrm{A} \cap \mathrm{C})=\phi \text { and }(\mathrm{B} \cap \mathrm{D})=\phi$ So $(A \cap C)$ and $(B \cap D)$ are always disjoint.

Hence, the correct answer is option: 'D'.

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