A method for computing the size of a set formed by a union of other sets, by subtracting the size of their intersections to avoid double-counting.
Basic counting principles: Counting the number of elements in a set using multiplication, addition, and subtraction principles.
Sets and subsets: Understanding the concept of sets, subsets, and operations such as intersection, union, and complement.
Venn diagrams: Construction and interpretation of Venn diagrams to analyze the relationship among sets.
Inclusion-Exclusion Principle: The principle that provides a formula for computing the size of the union of two or more sets without double-counting their intersecting elements.
Applications of Inclusion-Exclusion Principle: Using the principle to solve problems related to probability, combinatorics, and other fields.
Combinatorial probabilities: Understanding the concept of permutations, combinations, and binomial coefficients.
Conditional probabilities: Probability calculations by taking into account prior knowledge of the occurrence of certain events.
Probability distributions: Probability distributions such as binomial, Poisson, and normal distributions that are used to represent real-life scenarios.
Bayes' Theorem: A formula for computing the probability of an event based on prior knowledge or evidence.
Stirling's approximation: A mathematical formula that approximates factorial functions to make calculations easier.
Two Sets Inclusion-Exclusion Principle: This is the simplest form of the inclusion-exclusion principle, involving only two sets. It states that the size of the union of two sets is equal to the sum of the sizes of the individual sets minus the size of their intersection.
Three Sets Inclusion-Exclusion Principle: This principle is an extension of the two sets case to three sets. It states that the size of the union of three sets is equal to the sum of the sizes of the individual sets, minus the sum of the sizes of the pairwise intersections, plus the size of the intersection of all three sets.
Inclusion-Exclusion Principle for Indistinguishable Objects: This principle deals with the problem of counting the number of ways to select a certain number of objects from a given set without distinguishing between objects of the same type. It involves using the binomial coefficient and simple arithmetic.
Inclusion-Exclusion Principle for Distinguishable Objects: This principle is similar to the previous one, but it assumes that the objects are distinguishable. It involves using the principle of product and simple arithmetic.
Inclusion-Exclusion Principle for Probability: This principle deals with the probability of events in a given sample space. It states that the probability of at least one of a set of mutually exclusive events occurring is equal to the sum of their individual probabilities minus the sum of the probabilities of the pairwise intersections, plus the probability of their intersection.
Inclusion-Exclusion Principle for Multisets: This principle deals with multisets, which are sets that allow multiple instances of the same element. It involves using the binomial coefficient and simple arithmetic.
Generalized Inclusion-Exclusion Principle: This principle is a generalization of the previous principles to any number of sets. It involves using the principle of inclusion-exclusion recursively.