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Chapter 15: Problem 3

a) How many rows are needed to construct the (function) table for a Boolean function of \(n\) variables? b) How many different Boolean functions of \(n\) variables are there?

Short Answer

Expert verified
a) The function table for a Boolean function of \(n\) variables would need \(2^n\) rows. b) There are \(2^{2^n}\) different Boolean functions of \(n\) variables.

Step by step solution

01

Determine the number of rows in the Boolean function table

The number of rows in a Boolean function table is equivalent to all possible states that \(n\) variables can collectively have. Each variable can have two states: true (1) or false (0). So, for \(n\) variables, we will need \(2^n\) rows to represent all the possibilities.
02

Determine the number of different Boolean functions

For each of \(2^n\) rows in the Boolean function table, the output, or function value, can be either true (1) or false (0). Therefore, for each combination of variables, there are two possible outcomes. Thus, for \(2^n\) rows, there can be \(2^{2^n}\) different Boolean functions, as each row can independently choose one of the two outcomes.

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Most popular questions from this chapter

Obtain a minimal-product-of-sums representation for \(f(w, x, y, z)=\prod M(0,1,2,4,5,10,\), \(12,13,14)\).

a) If \(9 B_{1}, 9 A_{2}\) are Boolean algebras and \(f: 9 B_{1} \rightarrow S_{2}\) is one-to-one, onto, and such that \(f(x+y)=f(x)+f(y)\) and \(f(\bar{x})=\overline{f(x)}\), for all \(x, y \in \mathscr{B}_{1}\), prove that \(f\) is an isomorphism. b) State and prove another result comparable to that in part (a). (What principle is used here?)

In each of the following, \(f: B^{4} \rightarrow B\), where the Boolean variables (in order) are \(w, x, y\), and \(z\). Determine \(\left|f^{-1}(0)\right|\) and \(\left|f^{-1}(1)\right|\) if, as a minimal sum of products, \(f\) reduces to a) \(\bar{x}\) b) \(w y\) c) \(w \bar{y} z\) d) \(x+y\) e) \(x y+z\) f) \(x y \bar{z}+w\)

Let \(F_{6}\) denote the set of all Boolean functions \(f: B^{6} \rightarrow B\). a) What is \(\left|F_{6}\right|\) ? b) How many fundamental conjunctions (disjunctions) are there in \(F_{6}\) ? c) How many minterms (maxterms) are there in \(F_{6}\) ? d) How many functions \(f \in F_{6}\) have the value 1 when (exactly) two of its variables have the value 1 ? (In all other circumstances, the value of \(f\) may be 0 or 1 .) e) How many functions \(f \in F_{6}\) have the value 1 when at least two of its variables have the value 1? (In all other circumstances, the value of \(f\) may be 0 or 1.) f) Let \(u, v, w, x, y\), and \(z\) denote the six Boolean variables for the functions in \(F_{b-}\) How many of these functions are independent of \(x\) [that is, \(f(u, v, w, x, y, z)=\) \(f(u, v, w, \bar{x}, y, z)]\) ? How many are independent of all three of the Boolean variables \(x, y, z ?\)

Let \(a, b, c \in \mathscr{B}\), a Boolean algebra. Prove that \(a b+c=a(b+c)\) if and only if \(c \leq a\).
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