Question

From $10$ cards numbered $1$ to $10,$ a card is selected at random. consider the following events:
$A:$ The number on the card is a perfect square.
$B:$ The number is divisible by $3.$
$C:$ The number on the card is even.
$D:$ The number on the card is odd.
If $S$ is the sample space, then which of the following statement(s) is/are correct ?

• A. $A^c\cup B^c=S$
• B. $(A^c\cup B^c)\cup (A\cap B)=S$
• C. $C^c=D$
• D. $D^c=C$

Answer 1

Answer: (B), (C), (D)

$D^c=C$

Sample space $S=\{1,2,3,4,5,6,7,8,9,10\}$
Now,
$A:$ The number on the card is a perfect square.
$A=\{1,4,9\}$
$\Rightarrow A^c=S-A=\{2,3,5,6,7,8,10\}$
$B:$ The number is divisible by $3.$
$B=\{3,6,9\}$
$\Rightarrow B^c=\{1,2,4,5,7,8,10\}$
$C:$ The number on the card is even.
$C=\{2,4,6,8,10\}$
$\Rightarrow C^c=\{1,3,5,7,9\}$
$D:$ The number on the card is odd.
$D=\{1,3,5,7,9\}$
$\Rightarrow D^c=\{2,4,6,8,10\}$
$\Rightarrow A^c\cup B^c=\{1,2,3,4,5,6,7,8,10\}$
also $(A\cap B)=\left\{9\right\}$
and $(A^c\cup B^c)\cup (A\cap B)=\{1,2,3,4,5,6,7,8,9,10\}$