Search any question & find its solution
Question:
Answered & Verified by Expert
$\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ 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?
Options:
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?
Solution:
2397 Upvotes
Verified Answer
The correct answer is:
2 only
$\quad$ Let $\mathrm{A}=\{1,2\}$
$\mathrm{B}=\{3,4,0\}$
$\mathrm{C}=\{5,6,0\}$
$\mathrm{D}=\{7,8\}$
Such that $(\mathrm{A} \cap \mathrm{B})=(\mathrm{C} \cap \mathrm{D})=\phi$
$\Rightarrow(\mathrm{A} \cup \mathrm{C})=\{1,2,5,6,0\}$
$\Rightarrow(\mathrm{B} \cup \mathrm{D})=\{3,4,7,8,0\}$
$\Rightarrow(\mathrm{A} \cup \mathrm{C}) \cap(\mathrm{B} \cup \mathrm{D})=\{0\}$
So $(\mathrm{A} \cup \mathrm{C})$ and $(\mathrm{B} \cup \mathrm{D})$ are not always dispoint $\Rightarrow(\mathrm{A} \cap \mathrm{C})=\phi$ and $(\mathrm{B} \cap \mathrm{D})=\phi$
So $(\mathrm{A} \cap \mathrm{C})$ and $(\mathrm{B} \cap \mathrm{D})$ are always disjoint.
$\mathrm{B}=\{3,4,0\}$
$\mathrm{C}=\{5,6,0\}$
$\mathrm{D}=\{7,8\}$
Such that $(\mathrm{A} \cap \mathrm{B})=(\mathrm{C} \cap \mathrm{D})=\phi$
$\Rightarrow(\mathrm{A} \cup \mathrm{C})=\{1,2,5,6,0\}$
$\Rightarrow(\mathrm{B} \cup \mathrm{D})=\{3,4,7,8,0\}$
$\Rightarrow(\mathrm{A} \cup \mathrm{C}) \cap(\mathrm{B} \cup \mathrm{D})=\{0\}$
So $(\mathrm{A} \cup \mathrm{C})$ and $(\mathrm{B} \cup \mathrm{D})$ are not always dispoint $\Rightarrow(\mathrm{A} \cap \mathrm{C})=\phi$ and $(\mathrm{B} \cap \mathrm{D})=\phi$
So $(\mathrm{A} \cap \mathrm{C})$ and $(\mathrm{B} \cap \mathrm{D})$ are always disjoint.
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.