Problem 16 - Discrete Mathematics with Applications

Question

Construct a logic table for each boolean function defined by each boolean expression. $$x^{\prime} y z^{\prime}+x^{\prime}(y z)^{\prime}$$

Step-by-Step Solution

Answer

The logic table for the boolean function \(x^{\prime}yz^{\prime} + x^{\prime}(yz)^{\prime}\) is: | x | y | z | \(x^{\prime}yz^{\prime}\) | (yz)^{\prime} | \(x^{\prime}(yz)^{\prime}\) | F | |---|---|---|--------------|------------|---------------|---| | 0 | 0 | 0 | 0 | 1 | 1 | 1 | | 0 | 0 | 1 | 0 | 1 | 1 | 1 | | 0 | 1 | 0 | 1 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | 0 | 0 | 0 | | 1 | 0 | 0 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 0 | 0 | 0 | 0 | | 1 | 1 | 1 | 0 | 0 | 0 | 0 |

Step 1: Listing Input Combinations

Write down all possible input combinations for the variables x, y, and z. Since each variable can be either 0 or 1, there are 8 possible combinations. Step 2: Analyze the boolean expression

Step 2: Analyzing the Expression

Break the given boolean expression into two parts, the AND terms (product terms) and the OR terms (sum terms): - AND terms (product terms): \(x^{\prime}yz^{\prime}\) and \(x^{\prime}(yz)^{\prime}\) - OR Terms (sum terms): connect the two AND terms with the '+' operator Step 3: Evaluate the expression for each input combination

Step 3: Evaluating the Expression

For each input combination, evaluate the given boolean expression. This involves applying NOT (') to variables, computing the AND products, and then computing the OR-sum of the AND products. Step 4: Construct the logic table

Step 4: Constructing the Logic Table

Create a table with columns for each variable (x, y, z), the two AND terms, and the final output (F). Fill in the table with the input combinations and their corresponding output values obtained in Step 3. The completed logic table should look as follows: | x | y | z | \(x^{\prime}yz^{\prime}\) | (yz)^{\prime} | \(x^{\prime}(yz)^{\prime}\) | F | |---|---|---|--------------|------------|---------------|----------| | 0 | 0 | 0 | 0 | 1 | 1 | 1 | | 0 | 0 | 1 | 0 | 1 | 1 | 1 | | 0 | 1 | 0 | 1 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | 0 | 0 | 0 | | 1 | 0 | 0 | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 0 | 0 | 0 | 0 | | 1 | 1 | 1 | 0 | 0 | 0 | 0 | So, we have successfully constructed the logic table for the given boolean function.

Key Concepts

Boolean Function

A boolean function is like a building block for digital electronic systems, used widely in coding, computer hardware design, and even simple decision-making scenarios. It uses boolean algebra, where the values of the variables are true or false, often represented as 1 or 0, respectively. Boolean functions are mathematical representations of logic circuits and are composed of boolean operators like AND, OR, and NOT.
For instance, in the boolean function provided in the exercise, each of the variables (x, y, z) could represent a switch that is either ON (1) or OFF (0). The expression combines these using 'NOT' denoted by the prime (), 'AND' represented by concatenation, and 'OR' represented by the plus symbol (+). The goal is to evaluate how different combinations of these switch positions (inputs) affect the output (F).
In designing and analyzing digital circuits, boolean functions play a pivotal role. Each function corresponds to a specific circuit arrangement, and changing the function inevitably changes the circuit's behavior. It is, therefore, essential to understand the underlying boolean function when working with any digital logic system.

Boolean Expression Evaluation

Evaluating a boolean expression is a critical skill in various fields, including computer science and electrical engineering. It involves the systematic computation of the expression's value based on the boolean algebra rules.
To evaluate, we first consider the order of operations: NOT operations are performed first, followed by AND, and finally OR. In the given expression, \(x^{prime}yz^{prime}+x^{prime}(yz)^{prime}\), each part is resolved step by step. Parentheses dictate that operations within them take precedence. When evaluating, the NOT operation for variables (denoted by ') inverts their value. For instance, if is ON (1), \(x^{prime}\) is OFF (0). The AND operation results in ON only if all inputs are ON. The OR operation results in ON if at least one input is ON.
The evaluation of the boolean expression for each input combination results in the output value that forms part of the logic table. Understanding how to evaluate these expressions correctly allows us to predict the outcome of a digital circuit under any set of input conditions.

Truth Table

In the realm of boolean algebra, a truth table is an organized way of listing all possible combinations of inputs for a boolean function, along with the corresponding output values. Truth tables serve as a practical tool for visualizing and verifying the function of logic gates and circuits.
The logic table, or truth table, starts with columns for each input variable and ends with a column for the final output. Each row represents a unique combination of input values. For the given expression, the truth table summarizes the output of eight different combinations of x, y, and z. The process of constructing a truth table, as seen in the exercise, systematically determines the output for each combination.
Truth tables are not only helpful in the study and design of digital circuits, but they are also valuable in programming, philosophic logic, and systems design where binary alternatives are present. Mastery of truth tables and their constructions aids in error-checking, simplifying expressions, and understanding logical flow within a system.