1. **State the problem:** Simplify the expression $$\frac{1}{5} - \left\{ \frac{2}{5} + \frac{4}{15} \times \frac{1}{2} - \left( \frac{1}{2} + \frac{2}{5} \right) \right\}$$. 2. **Recall the order of operations:** Parentheses first, then multiplication and division, then addition and subtraction. 3. **Calculate inside the innermost parentheses:** $$\frac{1}{2} + \frac{2}{5} = \frac{5}{10} + \frac{4}{10} = \frac{9}{10}$$. 4. **Calculate the multiplication:** $$\frac{4}{15} \times \frac{1}{2} = \frac{4 \times 1}{15 \times 2} = \frac{4}{30} = \frac{2}{15}$$. 5. **Substitute back and simplify inside the braces:** $$\frac{2}{5} + \frac{2}{15} - \frac{9}{10}$$ 6. **Find a common denominator for $$\frac{2}{5}, \frac{2}{15}, \frac{9}{10}$$:** The least common denominator is 30. Convert each fraction: $$\frac{2}{5} = \frac{12}{30}, \quad \frac{2}{15} = \frac{4}{30}, \quad \frac{9}{10} = \frac{27}{30}$$. 7. **Perform the addition and subtraction:** $$\frac{12}{30} + \frac{4}{30} - \frac{27}{30} = \frac{16}{30} - \frac{27}{30} = \frac{16 - 27}{30} = \frac{-11}{30}$$. 8. **Substitute back into the original expression:** $$\frac{1}{5} - \left( \frac{-11}{30} \right) = \frac{1}{5} + \frac{11}{30}$$. 9. **Find common denominator for $$\frac{1}{5}$$ and $$\frac{11}{30}$$:** 30. Convert $$\frac{1}{5} = \frac{6}{30}$$. 10. **Add the fractions:** $$\frac{6}{30} + \frac{11}{30} = \frac{17}{30}$$. **Final answer:** $$\boxed{\frac{17}{30}}$$.