Question

(iii) A ∪ (B ∪ C) = (A ∪ B) ∪ (A ∩ C)

Answer 1

Given: A = {1,2,3,4}, B = {2,4,6,8}, C = {3,4,5,6} and U = $\left\{x|x\in N,x\in N,x<10\right\}$

We have to verify the following property:

$$\begin{gathered} \mathrm{A}\cap \left(\mathrm{B}\cup \mathrm{C}\right)=\left(\mathrm{A}\cap \mathrm{B}\right)\cup \left(\mathrm{A}\cap \mathrm{C}\right) \\ \mathrm{LHS}:\mathrm{A}\cap \left(\mathrm{B}\cup \mathrm{C}\right) \\ \mathrm{Here},\mathrm{B}\cup \mathrm{C}=\{2,3,4,5,6,8\} \\ \mathrm{Then}\mathrm{A}\cap \left(\mathrm{B}\cup \mathrm{C}\right)=\{2,3,4\} \\ \mathrm{RHS}:\left(\mathrm{A}\cap \mathrm{B}\right)\cup \left(\mathrm{A}\cap \mathrm{C}\right) \\ \mathrm{Here},\mathrm{A}\cap \mathrm{B}=\{2,4\}\mathrm{and}\mathrm{A}\cap \mathrm{C}=\{3,4\} \\ \mathrm{Then}\left(\mathrm{A}\cap \mathrm{B}\right)\cup \left(\mathrm{A}\cap \mathrm{C}\right)=\{2,3,4\} \\ \therefore \mathrm{LHS}=\mathrm{RHS} \\ \mathrm{Hence},\mathrm{A}\cap \left(\mathrm{B}\cup \mathrm{C}\right)=\left(\mathrm{A}\cap \mathrm{B}\right)\cup \left(\mathrm{A}\cap \mathrm{C}\right) \end{gathered}$$