Question

If $U= \{x : x \in N$ and $2 \le x \le 12\}$,
$A = \{x : x$ is an even prime$\}$, $B = \{x : x$ is a factor of $24\}$, then which of the following is not true?

• A. $A - B$ is an empty set
• B. $A - B = B \cap A'$
• C. $A' - B' = B - A$
• D. $(A \cap B)' = A' \cup B'$

Answer 1

Answer: (B)

$U= \{x : x \in N$ and $2 \le x \le 12\}$
$\Rightarrow U = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$
$A = \{x : x$ is an even prime$\}$
$\therefore A = \{2\}$
$B = \{x : x$ is a factor of $24\}$
$\therefore B = \{2, 3,4, 6, 8,12\}$
$\therefore A - B = \{2\} - \{2, 3, 4, 6, 8, 12\} = \phi$
$\therefore A - B$ is an empty set, is a true statement.
Also, $A - B = B \cap A'$ is not correct.
Again, $A' = \{3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$,
$B' = \{5, 7, 9, 10, 11\}$
$\therefore A' - B' = \{3, 4, 6, 8, 12\} = B - A$, which is true and $(A \cap B)' = A' \cup B'$ is the De Morgan’s law, which is also true.