Problem 32 - Discrete Mathematics with Applications

Question

Find the DNFs of the boolean functions $$\begin{array}{|ccc||c|} \hline x & y & z & f(x, y, z) \\ \hline 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The DNF (Disjunctive Normal Form) for the given Boolean function is: \(f(x, y, z) = (\overline{x} \cdot y \cdot z) \vee (x \cdot \overline{y} \cdot \overline{z}) \vee (x \cdot y \cdot \overline{z})\)

Step 1: Write conjunctions for each identified row

Create a conjunction for each row with an output of '1'. If the input variable is '0', the literal in the conjunction will be the negation of the variable; if it is '1', the literal will be the variable itself: - For row (0, 1, 1): \((\overline{x} \cdot y \cdot z)\), - For row (1, 0, 0): \((x \cdot \overline{y} \cdot \overline{z})\), - For row (1, 1, 0): \((x \cdot y \cdot \overline{z})\).

Step 2: Combine the conjunctions with OR

Combine the conjunctions that we got from step 1 using a disjunction (OR) operation, which will give us our DNF: \(f(x, y, z) = (\overline{x} \cdot y \cdot z) \vee (x \cdot \overline{y} \cdot \overline{z}) \vee (x \cdot y \cdot \overline{z})\) This is our final DNF for the given Boolean function.

Key Concepts

Disjunctive Normal Form

The Disjunctive Normal Form, often abbreviated as DNF, is a standard way of expressing a Boolean function. In this form, a Boolean expression is represented as a disjunction (logical OR) of several conjunctions (logical AND) of literals. A literal is either a variable or its negation. Each conjunction corresponds to a combination of variables that makes the function true (evaluates to '1').

To find the DNF of a Boolean function, you need to look at the truth table of the function. For each row where the output is '1', a conjunction of the input variables is generated. As such, each conjunction will include the variables in their true form if they are '1' in the row, and in negated form if they are '0'.

• This expression forms a sum of products, which is a name often interchangeable with DNF.
• It's useful because it gives a clear and simplified way to express any Boolean function.
• All logic combinations that result in a true output are explicitly defined in a DNF.

In the given exercise, the DNF for the Boolean function was derived by creating conjunctions for each row with a '1' output in the truth table and then combining those with OR operations.

Boolean Function

A Boolean Function is a function that delivers outputs based on logical operations on input values, where both inputs and outputs are either true or false (1 or 0). These functions are an essential part of digital logic design and computer science.

They often take the form of expressions involving logical variables and can be represented in various forms such as DNF, CNF (Conjunctive Normal Form), or as a truth table, which lists all possible combinations of inputs along with corresponding outputs.

• Each Boolean function is defined by its ability to output different combinations of 0's and 1's based on varying input conditions.
• Boolean functions are used to describe the logical operations and comparisons within circuits and computational logic.
• The logic behind these functions is implemented in hardware through gates and circuits.

The exercise presented involves deriving the DNF form by translating the truth table representation of a Boolean function into an algebraic expression that shows how different input combinations result in a logical '1'.

Logical Conjunction

Logical Conjunction, represented by the AND operator (\( \cdot \) or sometimes \( \land \)), is a fundamental operation in Boolean algebra that results in true if both operands are true, and false otherwise. In the context of DNF and Boolean functions, conjunctions are used to create product terms that cover instances when the function outputs true.

For instance, if variables x, y, and z represent different conditions, the conjunction \((x \cdot y \cdot z)\) will be true only when x, y, and z are all true. This rule applies even if one of the variables is negated.

• In DNF, conjunctions represent specific conditions that result in the output being true.
• Conjunctions are built using the truth values of the variables—variables are included in their true form if they are 1, and negated if they are 0, for the specific row.
• This logical operation is critical in forming the product terms that are summed together in the DNF expression.

In the provided solution, logical conjunctions from the truth table are combined to form the individual terms used in the DNF expression, based on rows that have a '1' output.

Logical Disjunction

Logical Disjunction is represented by the OR operator (\( \vee \) or sometimes \( \lor \)) and plays a crucial role in forming the overall structure of a Disjunctive Normal Form. This operation outputs true if at least one of the operands is true, making it versatile for combining conditions in Boolean expressions.

In DNF, the disjunction is used to connect different conjunctions that represent various scenarios when the Boolean function yields a true result.

• The essence of using disjunctions in DNF is to consolidate different groups of variable conditions that all lead to the same output.
• Without the OR operation, combining the multiple conjunctions to form the comprehensive truth of a Boolean function wouldn't be possible.
• It allows for an inclusive representation of different possibilities and their effects.

For the exercise at hand, once individual conjunctions are identified from the truth table, they are linked together using the OR operation to collectively represent the output function f(x, y, z) in its Disjunctive Normal Form.