Problem 1 - Discrete Mathematics with Applications

Question

When will the combinatorial circuit for each boolean expression produce 1 as the output? $$x^{\prime}$$

Step-by-Step Solution

Answer

The combinatorial circuit for the boolean expression \(x'\) will produce an output of 1 when the input variable x is 0.

Step 1: Understand the boolean expression x'

\(x'\) is the negation of the boolean variable x. The negation of a boolean variable inverts its value. If x is 0, then \(x'\) is 1, and if x is 1, then \(x'\) is 0.

Step 2: Check when x' is 1

Let's see when the expression \(x'\) has an output of 1. Since the expression is the negation of the variable x, it will be true (1) if x is false (0). In other words, if the value of x is 0, then the value of \(x'\) is 1. Therefore, the combinatorial circuit for the boolean expression \(x'\) will produce an output of 1 when the input variable x is 0.

Key Concepts

Combinatorial Circuit

A combinatorial circuit is a type of digital circuit whose output is purely based on the present input values. This means that unlike sequential circuits, combinatorial circuits do not rely on past input history. They work like a function in mathematics. For any given input, the combinatorial circuit provides one specific output. This is in contrast to sequential circuits that maintain a state based on historical inputs.

Combinatorial circuits are essential in computing for tasks like arithmetic computations, data processing, and logic operations. They employ logic gates like AND, OR, and NOT, which are the building blocks of these circuits.

• Simple design but effective for direct, immediate results.
• Used where quick output results are required based on present inputs.
• Do not have memory elements.

The combinatorial circuit in the exercise involves the negation operator, which flips the logic level of a boolean input, culminating in an output that is the exact opposite of the input.

Boolean Expression

Boolean expressions are algebraic expressions that result in a boolean value, either true (1) or false (0). They are built using boolean variables and logical operations such as AND, OR, and NOT.

In this exercise, the Boolean expression is denoted by \(x'\), which represents the negation or NOT operation on variable \(x\) .

• Boolean expressions like \(x'\) form the basis of digital logic design.
• They are used widely in decision-making processes in algorithms and electronic circuits.

The expression \(x'\) specifically negates the variable \(x\); if \(x\) is 0 (false), \(x'\) becomes 1 (true), and vice versa. Understanding this operation is crucial as it transforms information in various logic gates, circuits, and programming structures.

Negation

Negation is a fundamental operation in Boolean logic, symbolized by the prime sign (\( '\)). It plays a vital role in reversing the value of its operand. This concept is simple yet powerful—transforming true to false and false to true.

Negation is often implemented using a NOT gate in digital circuits. This gate takes an input and changes it to the opposite form, much like flipping a switch.

• An example of negation is \( x' \), which inverts \( x \).
• If \( x \) is 0, then \( x' \) is 1, highlighting its straightforward inverse operation.
• Essentially, negation helps in functions where decisions are based on opposite conditions.

In practical applications, negation aids in forming complex expressions and is pivotal for scenarios needing condition checks and logical flipping.