Question

Find the sets $A$ and $B$ if $\frac{A}{B}=\left\{1,5,7,8\right\},\frac{B}{A}=\left\{2,10\right\}$ and $A\cap B=\left\{3,6,9\right\}$. Are they unique?

Answer 1

Find the required sets.

Given: $\frac{A}{B}=\left\{1,5,7,8\right\},\frac{B}{A}=\left\{2,10\right\}$ and $A\cap B=\left\{3,6,9\right\}$

We know that $A=\left(\frac{A}{B}\right)\cup \left(A\cap B\right),B=\left(\frac{B}{A}\right)\cup \left(A\cap B\right)$

Thus,

$$\begin{gathered} A=\left\{1,5,7,8\right\}\cup \left\{3,6,9\right\} \\ \Rightarrow A=\left\{1,3,5,6,7,8,9\right\} \\ B=\left\{2,10\right\}\cup \left\{3,6,9\right\} \\ \Rightarrow B=\left\{2,3,6,9,10\right\} \end{gathered}$$

Since $A$ and $B$ have some common elements they are not unique.

Hence set $A=\left\{1,3,5,6,78,9\right\}$ and set $B=\left\{2,3,6,9,10\right\}$ and they are not unique.