Question

Let $\mathrm{n}\left(\mathrm{U}\right)=700,\mathrm{n}\left(\mathrm{A}\right)=200,\mathrm{n}\left(\mathrm{B}\right)=300$ and $\mathrm{n}(\mathrm{A}\cap \mathrm{B})=100$, then $n({\mathrm{A}}^{\mathrm{c}}\cap {\mathrm{B}}^{\mathrm{c}})=$

• A. $400$
• B. $600$
• C. $300$
• D. $200$

Answer 1

The correct option is C

$300$

Determine the value of $n({\mathrm{A}}^{\mathrm{c}}\cap {\mathrm{B}}^{\mathrm{c}})$

Let $\mathrm{U}$ denote the universal set, therefore the number of elements in the universal set,$\mathrm{n}\left(\mathrm{U}\right)=700$

Number of elements in the set $\mathrm{A}$, $\mathrm{n}\left(\mathrm{A}\right)=200$

Number of elements in the set $\mathrm{B}$, $\mathrm{n}\left(\mathrm{B}\right)=300$

Number of elements in $\mathrm{A}\cap \mathrm{B}$, $\mathrm{n}(\mathrm{A}\cap \mathrm{B})=100$

Now we know that,

$$\begin{gathered} {\mathrm{A}}^{\mathrm{c}}\cap {\mathrm{B}}^{\mathrm{c}}={\left(\mathrm{A}\cup \mathrm{B}\right)}^{\mathrm{c}} \\ \therefore \mathrm{n}({\mathrm{A}}^{\mathrm{c}}\cap {\mathrm{B}}^{\mathrm{c}})=\mathrm{n}{(\mathrm{A}\cup \mathrm{B})}^{\mathrm{c}} \\ =\mathrm{n}\left(\mathrm{U}\right)-\left\{\mathrm{n}(\mathrm{A}\cup \mathrm{B})\right\} \\ =\mathrm{n}\left(\mathrm{U}\right)-\left\{\mathrm{n}\left(\mathrm{A}\right)+\mathrm{n}\left(\mathrm{B}\right)-\mathrm{n}(\mathrm{A}\cap \mathrm{B})\right\} \\ =700-\left\{200+300-100\right\} \\ =700-400 \\ =300 \end{gathered}$$

Therefore the number of elements in ${\mathrm{A}}^{\mathrm{c}}\cap {\mathrm{B}}^{\mathrm{c}}=300$

Hence, option C is the correct answer.