Problem 14 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The Disjunctive Normal Form (DNF) of the boolean function \( f(x, y) = x \downarrow y \) is: $$ f(x, y) = (\bar{x} \land \bar{y}) \lor (\bar{x} \land y) \lor (x \land \bar{y}) $$

Step 1: Create the truth table

First, let's create the boolean function's truth table: | x | y | \(x \downarrow y\) | |:-:|:-:|:-------------:| | 0 | 0 | 1 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 0 |

Step 2: Identify the rows with output 1

Now we need to identify the rows with the output being true (1). These are: | x | y | \(x \downarrow y\) | |:-:|:-:|:-------------:| | 0 | 0 | 1 | | 0 | 1 | 1 | | 1 | 0 | 1 |

Step 3: Write the DNF

For each row with output 1, we create a product term. We use the variable if it is 1, and the negation of the variable if it is 0. Then we join all the product terms using the OR operation. With this in mind, the DNF is: $$ f(x, y) = (\bar{x} \land \bar{y}) \lor (\bar{x} \land y) \lor (x \land \bar{y}) $$

Key Concepts

Disjunctive Normal Form (DNF)

Disjunctive Normal Form, often abbreviated as DNF, is a way of structuring a Boolean expression. It involves writing the expression in a form that is a disjunction (OR operation) of multiple conjunctions (AND operations) of literals.

To better understand this, consider each conjunction (AND) as a term that corresponds to a row in a truth table where the output is true (1). In plain terms, DNF clearly lists all the scenarios under which the Boolean function evaluates to true.

For example, let's take a Boolean function expressed in terms of two variables, such as \( f(x, y) \). After analyzing which combinations of \( x \) and \( y \) return a true result, each of these combinations is transformed into a conjunctive clause. The entire function is the disjunction of these clauses, forming the DNF.

The example given shows the DNF for \( f(x, y) = x \downarrow y \), which results in \( (\bar{x} \land \bar{y}) \lor (\bar{x} \land y) \lor (x \land \bar{y}) \). This means the function is true when \( x \) and \( y \) are both false, \( x \) is false and \( y \) is true, or \( x \) is true and \( y \) is false.

Truth Table

The truth table is a fundamental component in understanding Boolean functions. It provides a visual way to display how the output of a Boolean function or operation depends on its inputs.

Each row of a truth table represents a possible combination of input values. For each combination, the corresponding outcome or result of the operation is shown, typically in binary form (with 0s and 1s).

When constructing a truth table for a Boolean function like \( f(x, y) \), the table will include all possible input combinations for \( x \) and \( y \), and the result of the function for each combination. This gives a comprehensive view of how an expression behaves under different input configurations. In our example, the truth table for \( x \downarrow y \) is provided, showing outputs resulting from each combination of \( x \) and \( y \).

Truth tables are instrumental in both analyzing Boolean functions and in determining expressions like the DNF, as they identify which rows lead to a true outcome.

Boolean Functions

Boolean functions form the building block of digital logic and computer science. They are algebraic expressions consisting of variables and operators that return binary output (true or false, or 1 or 0). Common operators include AND (\( \land \)), OR (\( \lor \)), NOT (\( \bar{} \)), as well as NAND, NOR, XOR, and XNOR.

Each Boolean function maps a set of binary inputs to a single binary output, providing a formalized way to describe logic gates and digital circuits. For instance, a function \( f(x, y) = x \downarrow y \) represents a NOR operation where the output is true only when both inputs are false.

By breaking down these functions into logical circuits or using truth tables, we can easily analyze and forecast their behavior across multiple scenarios. Through Boolean expressions, functions can be simplified or converted into formats like DNF for easier manipulation and implementation in designing circuits and solving logical problems.