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That is, if A is a subset of some set X, then if , and otherwise, where is one common notation for the indicator function; other common notations are , , [a] and ( ) . The indicator function of A is the Iverson bracket of the property of belonging to A ; that is,
The idea is similar: a set of random variables is k-wise independent if every subset of size k of those variables is independent. k-wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem MAXEkSAT.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
Equality between sets can be expressed in terms of subsets. Two sets are equal if and only if they contain each other: that is, A ⊆ B and B ⊆ A is equivalent to A = B. [30] [8] The empty set is a subset of every set: ∅ ⊆ A. [17] Examples: The set of all humans is a proper subset of the set of all mammals. {1, 3} ⊂ {1, 2, 3, 4}.
These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
Random variables are usually written in upper case Roman letters, such as or and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable.
Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. [1] A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph .
Every subset of an independent set is independent, i.e., for each , we have . This is sometimes called the hereditary property , or downward-closedness . Another term for an independence system is an abstract simplicial complex .