Switching and Finite Automata Theory by Zvi Kohavi

By Zvi Kohavi

This is often an important e-book in machine science,because the writer has written the entire mathematical fabrics in it.If you learn it,it can assist you to appreciate what "Algorithm" is.:)

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Additional info for Switching and Finite Automata Theory

Example text

Xn define a switching function. In other words, a switching function f (x1 , x2 , . . , xn ) is a correspondence that associates an element of the algebra with each of the 2n combinations of variables x1 , x2 , . . , xn . This correspondence is best specified by means of a truth table. Note that each truth table defines only one switching function, although this function may be expressed in a number of ways. The complement f (x1 , x2 , . . , xn ) is a function whose value is 1 whenever the value of f (x1 , x2 , .

It is often necessary to consider sets whose members are ordered pairs. Such a set of ordered pairs is called a binary relation. If R is a binary relation and the pair (a, b) is an element of R, we write a R b to indicate that a is related to b by R. We often specify relation R by the property that relates the members 26 Sets, relations, and lattices of each of its ordered pairs. For example, the binary relation “is less than” is denoted by a < b, “is equal to” is denoted by a = b, and so on. If A and B are two sets then the Cartesian product of A and B, denoted A × B, is the set containing all ordered pairs (a, b) such that a ∈ A and b ∈ B.

For example, the equivalence relation corresponding to Fig. 3 induces the partition π = {a, b, c; d, e; f }. It is quite obvious that the converse is also true and that every partition on S defines an equivalence relation in that set. A binary relation F in a set S of ordered pairs is called a function if and only if the existence of two pairs (a, b) and (a, c) in F such that their first coordinates are identical implies that b = c. In other words, a function is a set of ordered pairs in which no two pairs have the same first coordinate.

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