Problem 15 - Discrete Mathematics with Applications

Question

Find the DNF of each boolean function. $$f(x, y, z)=x \uparrow(y \uparrow z)$$

Step-by-Step Solution

Answer

The Disjunctive Normal Form of the given boolean function is: \(f(x, y, z) = x \cdot y \cdot z\).

Step 1: Analyzing the Boolean Function

We're given the boolean function \(f(x, y, z)=x \uparrow(y \uparrow z)\). The given function contains the "uptack" operation \(\uparrow\), which means that the function is using the Sheffer stroke (NAND) operation. In other terms, the function can be written as: $$f(x, y, z) = \overline{x \cdot (y \cdot z)}$$ where \(\overline{x}\) denotes the negation or NOT operation.

Step 2: Expanding the NAND gate expression

We begin by expanding the expression for y and z. Using the properties of NAND, we derive: $$y \cdot z \equiv \overline{y' + z'}$$ Now, substituting this into the overall function: $$f(x, y, z) = \overline{x \cdot (\overline{y' + z'})}$$

Step 3: Expanding the NAND gate expression for x

Next, we apply the properties of NAND to expand the expression involving x: $$x \cdot \overline{y' + z'} \equiv \overline{x' + (y' + z')}$$ Now, we have the given boolean function in the form of a negation over the sum: $$f(x, y, z) = \overline{x' + (y' + z')}$$

Step 4: Applying De Morgan's Law

To convert this expression into a disjunction of conjunctions, we apply De Morgan's Law: $$\overline{x' + (y' + z')} \equiv x'' \cdot (y'' \cdot z'')$$ Since double negations cancel out, the expression simplifies to: $$f(x, y, z) = x \cdot (y \cdot z)$$

Step 5: Convert to Disjunctive Normal Form

At this point, the simplified boolean function we have, \(f(x, y, z) = x \cdot (y \cdot z)\), is already in Disjunctive Normal Form, as it is a single conjunction: $$DNF \equiv x \cdot y \cdot z$$ In conclusion, the Disjunctive Normal Form of the given boolean function is: $$f(x, y, z) = x \cdot y \cdot z$$

Key Concepts

Boolean Function

A Boolean function is a mathematical expression built from binary variables, logical operators, and constants of 0 and 1. These functions produce results that are also binary, meaning they return either a 0 or a 1. For example, the Boolean function in our problem,
\(f(x, y, z) = x \uparrow (y \uparrow z)\), is constructed using the binary variables \(x, y\), and \(z\) and the NAND operation denoted by the Sheffer Stroke symbol \(\uparrow\).
In computational logic, Boolean functions are used to create logic gates, and in turn, entire circuits that make up the digital hardware of computers. Understanding how to manipulate and simplify these functions through various techniques such as finding their Disjunctive Normal Form (DNF) is crucial for tasks ranging from designing electronic circuits to programming in computer science.

Sheffer Stroke (NAND) Operation

The Sheffer Stroke, or NAND operation, is a fundamental building block in digital electronics. It is denoted by the symbol \(\uparrow\) and is defined as the negation of the conjunction of two variables. In other words, the expression
\(A \uparrow B\) is equivalent to \(\overline{A \cdot B}\), where \(\overline{A \cdot B}\) represents NOT (A AND B).

• \(1 \uparrow 0 = 1\)
• \(0 \uparrow 1 = 1\)
• \(0 \uparrow 0 = 1\)
• \(1 \uparrow 1 = 0\)

It is called a 'universal operation' because any Boolean function can be expressed using only NAND operations, as we could observe in the exercise. This property makes NAND gates very attractive for building digital circuits, allowing for the simplification of complex functions and the implementation of all other basic logic gates.

De Morgan's Law

De Morgan's Law is a pair of transformation rules that are pivotal in Boolean algebra and logic circuits. This law is used to simplify logical expressions and make connections between different forms of the same boolean formula. The two related rules are:

• The negation of a conjunction is the disjunction of the negations, expressed as \(\overline{A \cdot B} = \overline{A} + \overline{B}\).
• The negation of a disjunction is the conjunction of the negations, expressed as \(\overline{A + B} = \overline{A} \cdot \overline{B}\).

In the exercise, De Morgan's Law was applied to transform the negated disjunction into a conjunction, moving us closer to the goal of finding the Disjunctive Normal Form. Understanding De Morgan's Law is essential for any student tackling complex Boolean expressions, as it provides a reliable method for simplifying and rearranging terms.

Boolean Algebra

Boolean Algebra is the algebra of truth values and binary variables that make up the foundation of digital electronics and computer science logic. It operates with binary values (1 and 0), and it includes logical operations such as AND (conjunction), OR (disjunction), NOT (negation), NAND (Sheffer Stroke), NOR, XOR (exclusive OR), and XNOR (equivalence).
In the context of Boolean Algebra, the Disjunctive Normal Form (DNF) is one of the standard simplifications. A Boolean expression is in DNF if it is a disjunction of one or more conjunctions of literals. Here, a literal is either a variable or its negation. The exercise provided demonstrates a conversion of a Boolean function into DNF, highlighting the utility of Boolean Algebra in simplifying and standardizing the expression for practical applications such as programming logic and designing electrical circuits.