U of U ECE 6532 - Review of Probability Sets and Set Operations

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1Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilityObjectiveTo provide background material in support of topics in DigitalImage Processing that are based on probability and randomvariables.ReviewProbability & Random VariablesDigital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySets and Set OperationsProbability events are modeled as sets, so it is customary tobegin a study of probability by defining sets and some simpleoperations among sets.A set is a collection of objects, with each object in a set oftenreferred to as an element or member of the set. Familiarexamples include the set of all image processing books in theworld, the set of prime numbers, and the set of planetscircling the sun. Typically, sets are represented byuppercase letters, such as A, B, and C, and members of setsby lowercase letters, such as a, b, and c.Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySets and Set Operations (Con’t)We denote the fact that an element a belongs to set A byIf a is not an element of A, then we writeA set can be specified by listing all of its elements, or bylisting properties common to all elements. For example,suppose that I is the set of all integers. A set B consistingthe first five nonzero integers is specified using thenotationDigital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySets and Set Operations (Con’t)The set of all integers less than 10 is specified using thenotationwhich we read as "C is the set of integers such that eachmembers of the set is less than 10." The "such that" condition isdenoted by the symbol “ | “ . As shown in the previous twoequations, the elements of the set are enclosed by curly brackets.The set with no elements is called the empty or null set, denotedin this review by the symbol Ø.2Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySets and Set Operations (Con’t)Two sets A and B are said to be equal if and only if theycontain the same elements. Set equality is denoted byIf every element of B is also an element of A, we say that B isa subset of A:If the elements of two sets are not the same, we say that the setsare not equal, and denote this byDigital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySets and Set Operations (Con’t)Finally, we consider the concept of a universal set, which wedenote by U and define to be the set containing all elements ofinterest in a given situation. For example, in an experiment oftossing a coin, there are two possible (realistic) outcomes: headsor tails. If we denote heads by H and tails by T, the universal setin this case is {H,T}. Similarly, the universal set for theexperiment of throwing a single die has six possible outcomes,which normally are denoted by the face value of the die, so inthis case U = {1,2,3,4,5,6}. For obvious reasons, the universalset is frequently called the sample space, which we denote by S.It then follows that, for any set A, we assume that Ø ⊆ A ⊆ S,and for any element a, a ∈ S and a ∉ Ø.Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySome Basic Set OperationsThe operations on sets associated with basic probability theoryare straightforward. The union of two sets A and B, denotedbyis the set of elements that are either in A or in B, or in both. Inother words,Similarly, the intersection of sets A and B, denoted byis the set of elements common to both A and B; that is,Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySet Operations (Con’t)Two sets having no elements in common are said to be disjointor mutually exclusive, in which caseThe complement of set A is defined asClearly, (Ac)c=A. Sometimes the complement of A is denotedas .The difference of two sets A and B, denoted A − B, is the setof elements that belong to A, but not to B. In other words,3Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySet Operations (Con’t)It is easily verified thatThe union operation is applicable to multiple sets. Forexample the union of sets A1,A2,…,An is the set of points thatbelong to at least one of these sets. Similar comments applyto the intersection of multiple sets.The following table summarizes several important relationshipsbetween sets. Proofs for these relationships are found in mostbooks dealing with elementary set theory.Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySet Operations (Con’t)Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySet Operations (Con’t)It often is quite useful to represent sets and sets operations ina so-called Venn diagram, in which S is represented as arectangle, sets are represented as areas (typically circles), andpoints are associated with elements. The following exampleshows various uses of Venn diagrams.Example: The following figure shows various examples ofVenn diagrams. The shaded areas are the result (sets of points)of the operations indicated in the figure. The diagrams in the toprow are self explanatory. The diagrams in the bottom row areused to prove the validity of the expressionwhich is used in the proof of some probability relationships.Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilitySet Operations (Con’t)4Digital Image Processing, 3rd ed.www.ImageProcessingPlace.com© 1992–2008 R. C. Gonzalez & R. E. Woods Gonzalez & WoodsReview of ProbabilityRelative Frequency & ProbabilityA random experiment is an experiment in which it is notpossible to predict the outcome. Perhaps the best knownrandom experiment is


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U of U ECE 6532 - Review of Probability Sets and Set Operations

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