Question

Which of the following is correct?

1. $n(S\cup T)$ is maximum when $n(S\cap T)$ is least.

2. If $n\left(U\right)=1000,n\left(S\right)=720,n\left(T\right)=450$, then the least value of $n(S\cap T)=170$.

• A. Only $1$ is true
• B. Only $2$ is true
• C. Both $1$ and $2$ are true
• D. Both $1$ and $2$ are false

Answer 1

The correct option is C

Both $1$ and $2$ are true

Explanation for correct option

Using Union Formula,

$n(S\cup T)=n\left(S\right)+n\left(T\right)–n(S\cap T)$

If the intersection of two sets is minimum i.e., $n(S\cap T)$ then union of set is maximum $(S\cup T)$.

Hence, statement $1$ is true.

Given: $n\left(U\right)=1000,n\left(S\right)=720,n\left(T\right)=450$, Find $n(S\cap T)$

$$\begin{gathered} n(S\cap T)=n\left(S\right)+n\left(T\right)–n\left(SUT\right) \\ \Rightarrow n(S\cap T)=720+450-1000 \\ \Rightarrow n(S\cap T)=170 \end{gathered}$$

Hence statement $2$ is also true.
Hence the correct option is option (c).