Simplify the following expressions: a) (1/2 2/3) + (1/2 of 3/4) + 1/2 b) (2.4 1.6 3.4) / (0.8 3.2 * 5.1)

1. a)

Step 1: Simplify the terms in the brackets.
The expression is $(\frac{1}{2} \cdot \frac{2}{3}) + (\frac{1}{2} \text{ of } \frac{3}{4}) + \frac{1}{2}$.
"of" means multiplication.

$$\frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3}$$ $$\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$$

Step 2: Substitute the simplified terms back into the expression.

$$\frac{1}{3} + \frac{3}{8} + \frac{1}{2}$$

Step 3: Find a common denominator for 3, 8, and 2, which is 24. Convert each fraction.

$$\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$$ $$\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$$ $$\frac{1}{2} = \frac{1 \times 12}{2 \times 12} = \frac{12}{24}$$

Step 4: Add the fractions.

$$\frac{8}{24} + \frac{9}{24} + \frac{12}{24} = \frac{8 + 9 + 12}{24} = \frac{29}{24}$$

The simplified expression is $\boxed{\frac{29}{24}}$.

1. b)

Step 1: Rewrite the expression to simplify common factors.

$$\frac{2.4 \times 1.6 \times 3.4}{0.8 \times 3.2 \times 5.1} = \frac{2.4}{0.8} \times \frac{1.6}{3.2} \times \frac{3.4}{5.1}$$

Step 2: Simplify each fraction.

$$\frac{2.4}{0.8} = \frac{24}{8} = 3$$ $$\frac{1.6}{3.2} = \frac{16}{32} = \frac{1}{2}$$ $$\frac{3.4}{5.1} = \frac{34}{51} = \frac{2 \times 17}{3 \times 17} = \frac{2}{3}$$

Step 3: Multiply the simplified fractions.

$$3 \times \frac{1}{2} \times \frac{2}{3} = \frac{3 \times 1 \times 2}{2 \times 3} = \frac{6}{6} = 1$$

The simplified expression is $\boxed{1}$.

2. Given $\Sigma = \{1, 2, 3, 4, 5, 6, 7, 8, 9, ..., 20\}$.