Problem 10 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The truth table for the boolean expression \(x\left(y^{\prime} z+y z^{\prime}\right)\) is: | x | y | z | Output | |---|---|---|--------| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 0 | | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 0 |

Step 1: List all input combinations

First, list down all the possible input value combinations for x, y, and z. As there are 3 inputs, each being binary, there are a total of \(2^3 = 8\) combinations: | x | y | z | |---|---|---| | 0 | 0 | 0 | | 0 | 0 | 1 | | 0 | 1 | 0 | | 0 | 1 | 1 | | 1 | 0 | 0 | | 1 | 0 | 1 | | 1 | 1 | 0 | | 1 | 1 | 1 |

Step 2: Calculate the output value for each input combination

Now, calculate the output value for each input combination, using the given boolean expression: $ x\left(y^{\prime} z+y z^{\prime}\right) $ Recall the following boolean algebra properties: - \(x^{\prime}\) denotes the negation (NOT) of x, - \(x \cdot y\) denotes the conjunction (AND) of x and y, and - \(x+y\) denotes the disjunction (OR) of x and y. Compute the output value for each input combination: | x | y | z | Output | |---|---|---|--------| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 0 | | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 0 |

Step 3: Present the final truth table

Finally, present the truth table for the boolean expression \(x\left(y^{\prime} z+y z^{\prime}\right)\): | x | y | z | Output | |---|---|---|--------| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 0 | | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 0 |

Key Concepts

Truth Table

A truth table is a powerful tool in digital logic used to represent all possible values of a Boolean expression. It lists all combinations of the input variables and shows the corresponding output for each combination. This allows for a clear visualization of the logic. In the context of our exercise, we have three inputs: x, y, and z. Each of these inputs can be either 0 or 1, leading to a total of 8 combinations. Using these combinations, the truth table evaluates the Boolean expression:
\[ x(y' z + y z') \]
For each row in the table, we compute whether the expression is true or false (1 or 0). Truth tables are crucial because they provide an exhaustive list of outcomes, making them essential for error-checking in circuit design.

Boolean Algebra

Boolean algebra is the mathematical study of logic that uses variables to express conditions with two possible values: true or false, commonly represented by 1 and 0. It is the foundation of digital circuit design and computer programming. Some basic operations in Boolean algebra include:

• Negation (NOT): Denoted as \(x'\), it inverts the truth value. So, if \(x = 1\), then \(x' = 0\).
• Conjunction (AND): Denoted as \(x \cdot y\) or simply \(xy\), it results in 1 only if both inputs are 1. Otherwise, it results in 0.
• Disjunction (OR): Denoted as \(x + y\), it results in 1 if at least one input is 1. Otherwise, it results in 0.

In practice, Boolean algebra allows us to simplify complex logic circuits, making them more efficient. In our exercise, the expression \(y' z + y z'\) illustrates a mix of these operations, using both NOT and AND to determine the output when substituted into the larger expression.

Logic Gates

Logic gates are the building blocks of digital circuits. They perform basic logical functions that are fundamental to digital circuits. Each gate is designed to realize a fundamental Boolean operation:

• AND Gate: Implements the AND operation. It outputs 1 only if all its inputs are 1.
• OR Gate: Implements the OR operation. It outputs 1 if at least one of its inputs is 1.
• NOT Gate: Implements negation. It outputs the opposite of the input—it outputs 1 if the input is 0 and vice versa.

In combining these gates, we can create more complex circuits. For the exercise's expression \(x(y' z + y z')\), a combination of AND, OR, and NOT gates can be arranged to yield the desired output. Understanding logic gates helps us translate Boolean expressions into physical circuits that perform various computations and processes in machines.