In combinatorics, the addition principleШаблон:Sfn[1] or rule of sum[2][3] is a basic counting principle. Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways of doing another thing and we can not do both at the same time, then there are <math>A + B</math> ways to choose one of the actions.[2]Шаблон:Sfn In mathematical terms, the addition principle states that, for disjoint sets A and B, we have <math>|A\cup B| = |A| + |B|</math>.[1]
The addition principle can be extended to several sets. If <math>S_1, S_2,\ldots, S_n</math> are pairwise disjoint sets, then we have:Шаблон:Sfn[1]<math display="block">|S_1|+|S_2|+\cdots+|S_{n}| = |S_1 \cup S_2 \cup \cdots \cup S_n|. </math>This statement can be proven from the addition principle by induction on n.[1]
A person has decided to shop at one store today, either in the north part of town or the south part of town. If they visit the north part of town, they will shop at either a mall, a furniture store, or a jewelry store (3 ways). If they visit the south part of town then they will shop at either a clothing store or a shoe store (2 ways).
Thus there are <math>3 + 2 = 5</math> possible shops the person could end up shopping at today.
The inclusion–exclusion principle (also known as the sieve principleШаблон:Sfn) can be thought of as a generalization of the rule of sum in that it too enumerates the number of elements in the union of some sets (but does not require the sets to be disjoint). It states that if A1, ..., An are finite sets, thenШаблон:Sfn
<math display="block">\left|\bigcup_{i=1}^n A_i\right| =\sum_{i=1}^n\left|A_i\right|
-\sum_{i,j\,:\,1 \le i < j \le n}\left|A_i\cap A_j\right| + \sum_{i,j,k\,:\,1 \le i < j < k \le n}\left|A_i\cap A_j\cap A_k\right|-\ \cdots\ + (-1)^{n-1} \left|A_1\cap\cdots\cap A_n\right|.</math>
Subtraction principle
Similarly, for a given finite set S, and given another set A, if <math>A \subset S</math>, then <math>|A^c| = |S| - |A|</math>.[4][5] To prove this, notice that <math>|A^c| + |A| = |S|</math> by the addition principle.[5]
Applications
Шаблон:Expand section
The addition principle can be used to prove Pascal's rule combinatorially. To calculate <math>\binom{n + 1}{k}</math>, one can view it as the number of ways to choose k people from a room containing n children and 1 teacher. Then there are <math>\binom{n}{k}</math> ways to choose people without choosing the teacher, and <math>\binom{n}{k-1}</math> ways to choose people that includes the teacher. Thus <math>\binom{n+1}{k} = \binom{n}{k} + \binom{n}{k-1}</math>.[6]Шаблон:Rp
