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# 2.3 - 2-23 2.3 Functions 22 62 The union of two fuzzy sets...

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2-23 2.3 Functions 22 62. The union of two fuzzy sets S and T is the fuzzy set S T , where the degree of membership of an element in S T is the maximum of the degrees of membership of this element in S and in T . Find the fuzzy set F R of rich or famous people. 63. The intersection of two fuzzy sets S and T is the fuzzy set S T , where the degree of membership of an element in S T is the minimum of the degrees of membership of this element in S and in T . Find the fuzzy set F R of rich and famous people. 2.3 Functions Introduction In many instances we assign to each element of a set a particular element of a second set (which may be the same as the ﬁrst). For example, suppose that each student in a discrete mathematics class is assigned a letter grade from the set { A , B , C , D , F } . And suppose that the grades are A for Adams, C for Chou, B for Goodfriend, A for Rodriguez, and F for Stevens. This assignment of grades is illustrated in Figure 1. This assignment is an example of a function. The concept of a function is extremely impor- tant in mathematics and computer science. For example, in discrete mathematics functions are used in the deﬁnition of such discrete structures as sequences and strings. Functions are also used to represent how long it takes a computer to solve problems of a given size. Many computer programs and subroutines are designed to calculate values of functions. Recursive functions, which are functions deﬁned in terms of themselves, are used throughout computer science; they will be studied in Chapter 4. This section reviews the basic concepts involving functions needed in discrete mathematics. DEFINITION 1 Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A . We write f ( a ) = b if b is the unique element of B assigned by the function f to the element a of A .If f is a function from A to B , we write f : A B . Remark: Functions are sometimes also called mappings or transformations. Functions are speciﬁed in many different ways. Sometimes we explicitly state the assign- ments, as in Figure 1. Often we give a formula, such as f ( x ) = x + 1, to deﬁne a function. Other times we use a computer program to specify a function. Adams Chou Goodfriend Rodriguez Stevens A B C D F FIGURE 1 Assignment of Grades in a Discrete Mathematics Class.

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23 2 / Basic Structures: Sets, Functions, Sequences, and Sums 2-24 AB ab = f ( a ) f f FIGURE 2 The Function f Maps A to B . A function f : A B can also be deﬁned in terms of a relation from A to B . Recall from Section 2.1 that a relation from A to B is just a subset of A × B . A relation from A to B that contains one, and only one, ordered pair ( a , b ) for every element a A , deﬁnes a function f from A to B . This function is deﬁned by the assignment f ( a ) = b , where ( a , b ) is the unique ordered pair in the relation that has a as its ﬁrst element.
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2.3 - 2-23 2.3 Functions 22 62 The union of two fuzzy sets...

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