To solve this expression, let's break it down step-by-step.

The expression is:

[ \left{1 - \left[\left(1 - \frac{1}{4}\right) + \left(\frac{1}{2} - \frac{1}{3}\right)\right]\right} - \left{1 - \left[\left(\frac{1}{3} - \frac{1}{4}\right) - \left(\frac{1}{5} - \frac{1}{6}\right)\right]\right} ]

Step 1: Solve the Inner Brackets

First Bracket: $\left(1 - \frac{1}{4}\right) + \left(\frac{1}{2} - \frac{1}{3}\right)$

1. Calculate $1 - \frac{1}{4}$: $1 - \frac{1}{4} = \frac{4 - 1}{4} = \frac{3}{4}$

2. Calculate $\frac{1}{2} - \frac{1}{3}$: $\frac{1}{2} - \frac{1}{3} = \frac{3 - 2}{6} = \frac{1}{6}$

3. Now combine these results: $\frac{3}{4} + \frac{1}{6} = \frac{9 + 2}{12} = \frac{11}{12}$

So, the value of the first bracket is $\frac{11}{12}$.

Second Bracket: $\left(\frac{1}{3} - \frac{1}{4}\right) - \left(\frac{1}{5} - \frac{1}{6}\right)$

1. Calculate $\frac{1}{3} - \frac{1}{4}$: $\frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12}$

2. Calculate $\frac{1}{5} - \frac{1}{6}$: $\frac{1}{5} - \frac{1}{6} = \frac{6 - 5}{30} = \frac{1}{30}$

3. Now combine these results: $\frac{1}{12} - \frac{1}{30} = \frac{5 - 2}{60} = \frac{3}{60} = \frac{1}{20}$

So, the value of the second bracket is $\frac{1}{20}$.

Step 2: Substitute Back and Solve

Now we substitute these values into the original expression:

$\left\{1 - \frac{11}{12}\right\} - \left\{1 - \frac{1}{20}\right\}$

1. Calculate $1 - \frac{11}{12}$: $1 - \frac{11}{12} = \frac{12 - 11}{12} = \frac{1}{12}$

2. Calculate $1 - \frac{1}{20}$: $1 - \frac{1}{20} = \frac{20 - 1}{20} = \frac{19}{20}$

3. Now combine these results: $\frac{1}{12} - \frac{19}{20} = \frac{5 - 57}{60} = -\frac{52}{60} = -\frac{13}{15}$

Final Answer

$-\frac{13}{15}$

Would you like more details or have questions on any part of this solution?

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Here are some related questions you might consider:

1. How do we simplify complex fractions in nested brackets?
2. Why do we find a common denominator when subtracting fractions?
3. How would the expression change if different numbers were used?
4. What are other methods to verify the answer?
5. How do we convert the fraction result to a decimal?

Tip: Always simplify fractions at each step to make calculations easier and avoid mistakes.