Question

let A={1,2,4,5} B={2,3,5,6} C={4,5,6,7}. verify the following identity
A intersection (B▲C)=(A intersection B)▲ (A intersection C).

Answer 1

$$\begin{gathered} \mathrm{A}=\left\{1,2,4,5\right\},\mathrm{B}=\left\{2,3,5,6\right\}\mathrm{and}\mathrm{C}=\left\{4,5,6,7\right\} \\ \left(\mathrm{B}∆\mathrm{C}\right)=\left[\left(\mathrm{B}-\mathrm{C}\right)\cup \left(\mathrm{C}-\mathrm{B}\right)\right] \\ =\left[\left\{2,3\right\}\cup \left\{4,7\right\}\right]=\left\{2,3,4,7\right\} \\ \mathrm{A}\cap \mathrm{B}=\left\{2,5\right\} \\ \mathrm{A}\cap \mathrm{C}=\left\{4,5\right\} \\ \mathrm{LHS} \\ \mathrm{A}\cap \left(\mathrm{B}∆\mathrm{C}\right)=\left\{1,2,4,5\right\}\cap \left\{2,3,4,7\right\} \\ =\left\{2,4\right\} \\ \mathrm{RHS} \\ \left(\mathrm{A}\cap \mathrm{B}\right)∆\left(\mathrm{A}\cap \mathrm{C}\right)=\left\{2,5\right\}∆\left\{4,5\right\} \\ =\left(\left\{2,5\right\}-\left\{4,5\right\}\right)\cup \left(\left\{4,5\right\}-\left\{2,5\right\}\right) \\ =\left\{2\right\}\cup \left\{4\right\}=\left\{2,4\right\} \\ \mathrm{LHS}=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{Verified}. \end{gathered}$$