Basic Theory of Logic Functions:Basic Theorems

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Basic Theorems

Theory on logic functions where the values of variables and their functions are 0 or 1 only is called switching theory. Here, let us discuss the basics of switching theory.

Let us denote the set of input variables by the vector expression (x1, x2, …, xn). There are 2n different input vectors when each of these n variables assume the value 1 or 0. An input vector (x1, x2, …, xn) such that f (x1, x2, …, xn) = 1 or 0 is called a true (input) vector, or a false (input) vector of f, respectively. Vectors with n components are often called n-dimensional vectors if we want to emphasize that there are n components. When the value of a logic function f is specified for each of the 2n vectors (i.e., for every combination of the values of x1, x2, …, xn), f is said to be completely specified. Otherwise, f is said to be incompletely specified; that is, the value of f is specified for fewer than 2n vectors. Input vectors for which the value of f is not specified are called don’t-care conditions usually denoted by “d” or “*,” as described in Chapter 23. These input vectors are never applied to a network whose output realizes f, or the values of f for these input vectors are not important. Thus, the corresponding values of f need not be considered.

If there exists a pair of input vectors (x1, …, xi – 1, 0, xi+1, …, xn) and (x1, …, xi–1, 1, xi+1, …, xn) that differ only in a particular variable xi, such that the values of f for these two vectors differ, the logic function f(x1, x2, …, xn) is said to be dependent on xi. Otherwise, it is said to be independent of xi. In this case, f can be expressed without the xi in the logic expression of f. If f is independent of xi, xi is called a dummy variable. If f(x1, x2, …, xn) depends on all its variables, it is said to be non-degenerate; otherwise, degenerate.

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Then we can further expand with respect to x3. And so on.

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Basic Theory of Logic Functions-0344

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