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 of the given boolean function \(f(x, y) = x \downarrow y\) is: \(f(x, y) = (\overline{x} \land \overline{y}) \lor (\overline{x} \land y) \lor (x \land \overline{y})\)

Step 1: Create Truth Table

First, let's create a truth table for the given function. The truth table will illustrate all possible combinations of input values and their corresponding output for the function \(f(x, y) = x \downarrow y\). x | y | f(x, y) ---|-----|--------- 0 | 0 | 1 0 | 1 | 1 1 | 0 | 1 1 | 1 | 0 The truth table is complete.

Step 2: Identify True Rows

Now, let's identify the rows where the function is true (equal to 1): - f(0,0) = 1 - f(0,1) = 1 - f(1,0) = 1 We will disjunctively join these cases using the OR operation to form the Disjunctive Normal Form.

Step 3: Express the DNF

To express the Disjunctive Normal Form, we will join the cases found in step 2 using OR: \(f(x, y) = (\overline{x} \land \overline{y}) \lor (\overline{x} \land y) \lor (x \land \overline{y})\) This is the Disjunctive Normal Form for the given boolean function \(f(x, y) = x \downarrow y\).

Key Concepts

Boolean Function

A Boolean function is a mathematical expression composed of binary variables, logical operators, and binary outcomes. In essence, it provides a systematic way of representing and analyzing logical statements. The Boolean function operates on binary values, typically represented as 1 (True) and 0 (False), and returns a single binary outcome. Unlike typical algebraic functions that deal with a wide range of numbers, Boolean functions are fundamentally used in digital circuits and computer algorithms to manage decision-making processes and logical operations.

For example, in the exercise provided, the Boolean function is defined as \(f(x, y) = x \downarrow y\), where \(\downarrow\) represents the NOR logical operator. It demonstrates how to extract information from a given Boolean expression and translate it into a form that can be easily implemented in electrical circuits or computer code. Understanding the interplay of the logical operators within a Boolean function is essential for solving complex problems in computer science and engineering.

Truth Table

A truth table serves as a powerful tool to display all possible values of a Boolean function. It enables us to visualize the function's outputs for each combination of inputs, solidifying our understanding of the function's behavior. Truth tables are particularly useful when dealing with Boolean logic, as they present a clear, structured way to deduce the logical outcome of complex expressions.

In the context of the given exercise, the truth table is constructed to show all the possible outputs of the function \(f(x, y) = x \downarrow y\) with \(x\) and \(y\) having the binary values 0 and 1. This table facilitates the discovery of patterns or the derivation of a certain form of the function, such as the Disjunctive Normal Form. Once the truth table is filled, it becomes straightforward to identify when the function's output will be True (1) and use those instances to build the desired logical expression.

Logical Operators

Logical operators are the building blocks of Boolean logic that dictate the rules of interaction between different Boolean variables. They include AND (\(\land\)), OR (\(\lor\)), NOT (\(\overline{x}\)), NAND (\(\uparrow\)), NOR (\(\downarrow\)), XOR (exclusive OR), and XNOR (exclusive NOR). Each operator has a specific set of rules that determines the output based on the inputs.

In the exercise provided, the NOR operator (\(\downarrow\)) is used. This operator gives a result of 1 (True) only when all the operands are 0 (False). The expression \(f(x, y) = x \downarrow y\) returns a True only when both \(x\) and \(y\) are false. The step-by-step solution elucidates how these operators can be combined into a Disjunctive Normal Form (DNF), which is a standardized way to express Boolean functions as a disjunction (OR) of conjunctions (AND) of literals (variables or their negation). The correct use and understanding of logical operators are critical for analyzing and simplifying Boolean expressions in computational logic design.