To solve the expression \(2 - \frac{12}{19} \div \frac{13}{12}\), we need to follow the order of operations (PEMDAS/BODMAS). This means we handle the division before the subtraction.

First, let's simplify the division part:

\[
\frac{12}{19} \div \frac{13}{12}
\]

Dividing by a fraction is the same as multiplying by its reciprocal:

\[
\frac{12}{19} \div \frac{13}{12} = \frac{12}{19} \times \frac{12}{13}
\]

Now, multiply the fractions:

\[
\frac{12 \times 12}{19 \times 13} = \frac{144}{247}
\]

So, the expression now becomes:

\[
2 - \frac{144}{247}
\]

To subtract these, we need a common denominator. The number 2 can be written as a fraction with the same denominator:

\[
2 = \frac{2 \times 247}{247} = \frac{494}{247}
\]

Now, subtract the fractions:

\[
\frac{494}{247} - \frac{144}{247} = \frac{494 - 144}{247} = \frac{350}{247}
\]

Thus, the simplified result is:

\[
\frac{350}{247}
\]

This fraction can be simplified further by finding the greatest common divisor (GCD) of 350 and 247. The GCD of 350 and 247 is 1, so the fraction is already in its simplest form.

Therefore, the final answer is:

\[
\frac{350}{247}
\]