By A. T. Berztiss and Werner Rheinboldt (Auth.)

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**Example text**

Consider the first row. The entries for / and g are given by the definitions above. 3. From the table it is seen t h a t / © # = # , / * # = / , / ' = s, g' = p. 7 we have defined functional equality. 7 are in fact functions. For example, we have to show that i f / ® g = h a n d / © # = h , then h = h . 4 Let (B, ©, *, 0, 1> be a Boolean algebra. Let/and g denote Boolean functions on B into B. Then THEOREM n (0 (ii) f®9> / * 9, (iii) / ' are also functions. 2b. )(a) = c . 7, (f®g)(a) = f(a) ®g(a).

Therefore, strictly for the sake of convenience, we first define a symbol designating the number of elements in a finite set. In what then follows we shall deal exclusively with finite nonempty sets. 18 Let A be a finite set. Then \A\ denotes the number of elements in A. DEFINITION Example A = {1, 2, 2, 5}. Here \A\ = 3. 19 Let A be a set with \A\ = n. , a y e A then this ordered m-tuple is an m-sample of A. If all a in an ra-sample are distinct, the m-sample is an m-permutation of A. In particular, an w-permutation is called simply a permutation of A.

5. 6. 1. 2 0 and 1 are Boolean forms. A variable x (i = 1, 2, . . , n) is a Boolean form. If a is a Boolean form, then so is (a). If a is a Boolean form, then so is a'. If a and /? are Boolean forms, then so is a ©/?. If a and fi are Boolean forms, then so is a * /?. Only expressions given by Statements 1-6 are Boolean forms. t 2 Functions and Relations 50 Example Consider the expression (x ®x )'@(x[*x ). Let us determine whether or not it is a Boolean form. This is a recognition problem. We find successively that x x , x ®x , (x ®x ) (x ®x )\ x[ x[*x , (x[*x ) Boolean forms.