$|\mathcal{P}(S)| = 2^{|S|}$ $\text{where } \mathcal{P}(S) = \text{the power set (set of all subsets) of } S$ Discrete Mathematics is the branch of mathematics concerned with countable, distinct structures — including sets, logic, combinatorics, graph theory, and number theory — as opposed to continuous quantities studied in calculus. In probability and actuarial science, discrete mathematics provides the language and tools for counting outcomes (combinatorics), defining events (set theory), and reasoning about logical relationships between them. > [!example]- Counting Subsets of a Risk Portfolio {💡 Example} > An insurer has 4 distinct risk categories: Fire, Flood, Theft, and Liability. How many distinct subsets of these risks could be included in a policy? > > > [!answer]- Answer > > The number of subsets of a set with $|S| = 4$ elements is: > > $|\mathcal{P}(S)| = 2^4 = 16$ > > This includes the empty set (no coverage) and the full set (all four risks covered), giving 16 possible coverage combinations.