Question

Let A={1,2,3,4}, B={2,4,6} . Then the number of sets of C such that A intersection B is a subset of C is a subset of A union B is

Answer 1

Hi,
A set B is a subset of a set A if all element of B are also element of A. Such a relation between sets is denoted by B ⊆ A

$$\begin{gathered} \mathrm{Here} \\ \mathrm{A}=\left\{1,2,3,4\right\} \\ \mathrm{B}=\left\{2,4,6\right\} \\ \mathrm{Now}\left(\mathrm{A}\cap \mathrm{B}\right)\subseteq \mathrm{C}\subseteq \left(\mathrm{A}\cup \mathrm{B}\right) \\ \mathrm{So}\mathrm{A}\cap \mathrm{B}=\left\{2,4\right\} \\ \mathrm{A}\cup \mathrm{B}=\left\{1,2,3,4,6\right\} \\ \mathrm{so}\mathrm{here} \\ \left\{2,4\right\}\subseteq \mathrm{C}\subseteq \left\{1,2,3,4,6\right\} \\ \mathrm{now}\mathrm{this}\mathrm{condition}\mathrm{implies}\mathrm{that}\mathrm{set}\mathrm{C}\mathrm{must}\mathrm{have}2,4,\mathrm{as}\mathrm{its}\mathrm{element} \\ \mathrm{and}\mathrm{it}\mathrm{is}\mathrm{also}\mathrm{a}\mathrm{subset}\mathrm{of}\mathrm{set}\left\{1,2,3,4,6\right\} \\ \mathrm{So}\mathrm{C}\mathrm{can}\mathrm{be}\left\{\right\}or\varphi ,\left\{2,4\right\},\left\{1,2,4\right\},\left\{3,2,4\right\},\left\{6,2,4\right\},\left\{1,2,4,3\right\},\left\{1,2,4,6\right\}, \\ \left\{1,2,4,3,6\right\} \\ \mathrm{so}\mathrm{there}\mathrm{are}8\mathrm{possibilities} \end{gathered}$$