# The Addition and Subtraction Principles

Recall that if $A$ is a set and $A_1, A_2, ..., A_n \subset A$ such that $A = \bigcup_{i=1}^{n} A_i$ and for $j \neq k$ we have that $A_j \cap A_k = \emptyset$ for each $j, k = 1, 2, ..., n$ then the collection of sets $A_1, A_2, ..., A_n$ form a partition of $A$

We are now ready to look at two relatively simple principles known as the **Addition** principle and **Subtraction** principle which seem rather obvious, but we still go through the process of definition them below.

## The Addition Principle

Definition (The Addition Principle): If $A_1, A_2, ..., A_n \subset A$ form a partition of the finite set $A$ then the size of $A$ is equal to the sum of the sizes of $A_1, A_2, ..., A_n$, that is $\lvert A \rvert = \lvert A_1 \rvert + \lvert A_2 \rvert + ... + \lvert A_n \rvert = \sum_{i=1}^{n} \lvert A_i \rvert$. |

For example, consider the set $A = \{ x, y, z \}$. If $A_1, A_2 \subset A$ are defined by $A_1 = \{ x \}$ and $A_2 = \{ y, z \}$. Clearly $A_1$ and $A_2$ form a partition of $A$ and we can clearly see that:

(1)Furthermore:

(2)Therefore we have that $\lvert A \rvert = \lvert A_1 \rvert + \lvert A_2 \rvert$.

## The Subtraction Principle

Recall that if $A$ is a set and $B \subset A$ then the set theoretic difference of $A$ minus $B$ is defined as:

(3)We are now ready to define the subtraction principle.

Definition (The Subtraction Principle): If $A$ is a finite set and $B \subseteq A$ then the number of elements in $A \setminus B$ is equal to the number of elements in set $A$ minus the number of elements in set $B$, that is $\lvert A \setminus B \rvert = \lvert A \rvert - \lvert B \rvert$. |

*Note that if $A$ is a finite set and $B \subseteq A$ then $B$ is by extension a finite set too and so the formula given in the definition above is well-defined.*

Consider the sets $A$ and $B \subset A$ defined as:

(4)and:

(5)We have that $\lvert A \rvert = 100$ and $\lvert B \rvert = 50$. Furthermore we note that:

(6)Therefore $\lvert A \setminus B \rvert = 50$. As we can see, $\lvert A \setminus B \rvert = \lvert A \rvert - \lvert B \rvert$.