additionsprincipen

Additionsprincipen, or the addition principle, is a basic rule in combinatorics and discrete mathematics that counts the size of a union of mutually exclusive options. If a task can be performed in a ways for option A and in b ways for option B, and the options cannot occur simultaneously, then there are a+b total ways to perform the task.

Formal statement: Let A and B be finite sets representing distinct outcome sets with A and B disjoint (A ∩ B = ∅). Then the size of their union satisfies |A ∪ B| = |A| + |B|. More generally, for a finite collection of pairwise disjoint sets A1, A2, ..., An, the size of their union equals the sum of their sizes: |∪i Ai| = Σi |Ai|.

Relation to other counting rules: The addition principle complements the multiplication principle. The multiplication principle handles sequential choices within a single path (or the combination of independent steps), while the addition principle handles alternatives that cannot occur together.

Examples: If a vending machine offers 3 types of coffee and 2 types of tea, there are 3+2 = 5 possible hot beverages. If a restaurant menu has 4 vegetarian dishes and 5 meat dishes, there are 9 total main-course options.

Probability variant: If events A and B are disjoint, P(A ∪ B) = P(A) + P(B). If they are not disjoint, inclusion-exclusion must be used: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Limitations: The principle requires the options to be disjoint. When overlap exists, one must subtract the intersection (or use the general inclusion-exclusion principle).