3.1 The sum rule

The sum rule applies when there are alternative ways to do the same thing, or different outcomes from the same action.

For example, let us consider the number of rooms to search in a building with three floors. One of the rooms has an item that we want o find. Assume the first floor has 20 rooms, the second floor has 15 rooms, and the third floor has 10 rooms. The total number of rooms to search is, then $20+15+10=45$ .

The sum rule is linked with disjunction (``or''). In our previous example, the item can be in a room in the first floor, or a room in the second floor, or a room in the third floor. Another way look at this is that the sum rule deals with alternatives.

In more formal terms, let the set $X_i$ represent the possibilities offered by option $i$ . Let us assume there are $n$ options. Then we assume that all the options are disjoint, which means $\forall i,j \in [1\ldots n]: (i \ne j) \Rightarrow X_i \cap X_j = \{\}$ . Then the set of all possibilities from all the options is $\bigcup_{i=1}^{n} X_i$ .

Because the possibilities of each option is disjoint from that of the other options, $\vert\bigcup_{i=1}^{n} X_i\vert = \sum_{i=1}^{n} \vert X_i\vert$ .