Question

In a hospital, $89\%$ has the disease $A$ and $98\%$ has the disease $B$, $K\%$ is the number of patients who has both, then $K$ can't be the subset of

Answer 1

Determine the minimum and maximum value of $K$

Let us assume that the total number of patients in the hospital is $n\left(T\right)=100$.

It is given that $89\%$ has the disease $A$, $98\%$ has the disease $B$ and $K\%$ is the number of patients who has both.

Thus, the number of patients who have the disease $A$, $n\left(A\right)=89$.

The number of patients who have the disease $B$, $n\left(B\right)=98$.

The number of patients who have both the disease $A$ and the disease $B$, $n\left(A\cap B\right)=K$.

Hence, $\max \left[n\left(A\right),n\left(B\right)\right]=\mathrm{n}\left(\mathrm{B}\right)=98$.

It is known that $\max \left[n\left(A\right),n\left(B\right)\right]\le n\left(A\cup B\right)\le n\left(T\right)$.

$\Rightarrow 98\le n\left(A\cup B\right)\le 100$.

It is also known that $n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)$.

So, $98\le n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)\le 100$.

$$\begin{gathered} \Rightarrow 98\le 89+98-n\left(A\cap B\right)\le 100 \\ \Rightarrow 98\le 187-n\left(A\cap B\right)\le 100 \\ \Rightarrow 98-187\le -n\left(A\cap B\right)\le 100-187 \\ \Rightarrow -89\le -n\left(A\cap B\right)\le -87 \\ \Rightarrow 89\ge n\left(A\cap B\right)\ge 87 \end{gathered}$$

Thus, $n\left(A\cap B\right)\in \left[87,89\right]$.

Hence, $n\left(A\cap B\right)$ is not a subset of $\left\{x:x\in \mathrm{ℝ}-\left[87,89\right]\right\}$.