Let's simplify each expression step-by-step.

### a) \(\frac{2x - 1}{x - 1} - \frac{x - 2}{x - 1}\)

Since the denominators are the same, we can combine the fractions:

\[
\frac{(2x - 1) - (x - 2)}{x - 1}
\]

Now simplify the numerator:

\[
2x - 1 - x + 2 = x + 1
\]

Thus, we have:

\[
\frac{x + 1}{x - 1}
\]

### b) \(\frac{2(2xy - 1)}{3x^2y^2} - \frac{xy - 2}{3x^2y^2}\)

Again, the denominators are the same:

\[
\frac{2(2xy - 1) - (xy - 2)}{3x^2y^2}
\]

Now simplify the numerator:

\[
2(2xy - 1) - (xy - 2) = 4xy - 2 - xy + 2 = 3xy
\]

So we have:

\[
\frac{3xy}{3x^2y^2} = \frac{xy}{x^2y^2} = \frac{1}{xy}
\]

### c) \(\frac{3x - 2}{x + 1} - \frac{2x - 1}{x + 1}\)

Combine the fractions:

\[
\frac{(3x - 2) - (2x - 1)}{x + 1}
\]

Simplify the numerator:

\[
3x - 2 - 2x + 1 = x - 1
\]

Thus, we have:

\[
\frac{x - 1}{x + 1}
\]

### d) \(\frac{2x - 1}{x - 1} - \frac{x - 2}{x - 1}\)

This is similar to part a and can be simplified the same way:

\[
\frac{(2x - 1) - (x - 2)}{x - 1} = \frac{x + 1}{x - 1}
\]

### e) \(\frac{2(2xy - 1)}{3x^2y^2} - \frac{xy - 2}{3x^2y^2}\)

This is similar to part b and simplifies to:

\[
\frac{3xy}{3x^2y^2} = \frac{xy}{x^2y^2} = \frac{1}{xy}
\]

### f) \(\frac{3x - 2}{x + 1} - \frac{2x - 1}{x + 1}\)

This is similar to part c:

\[
\frac{(3x - 2) - (2x - 1)}{x + 1} = \frac{x - 1}{x + 1}
\]

### Summary of Results:

- a) \(\frac{x + 1}{x - 1}\)
- b) \(\frac{1}{xy}\)
- c) \(\frac{x - 1}{x + 1}\)
- d) \(\frac{x + 1}{x - 1}\)
- e) \(\frac{1}{xy}\)
- f) \(\frac{x - 1}{x + 1}\)