Question

For three sets A, B and C, show that
(i) need not imply B = C.
(ii)

Answer 1

(i) Let A = {2, 4, 5, 6}, B = {6, 7, 8, 9} and C = {6, 10, 11, 12,13}

$$\begin{gathered} \mathrm{So},A\cap B=\left\{6\right\}\mathrm{and}A\cap C=\left\{6\right\} \\ \mathrm{Hence},A\cap B=A\cap C\mathrm{but}B\ne C \end{gathered}$$

(ii)

$$\begin{gathered} \mathrm{Let}z\in C-B...\left(1\right) \\ \Rightarrow z\in C\mathrm{and}z\notin B \\ \Rightarrow z\in C\mathrm{and}z\notin A\left[\because A\subset B\right] \\ \Rightarrow z\in C-A...\left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get} \\ C-B\subset C-A \end{gathered}$$