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Chapter 13: Problem 2

Write the following Boolean expression in the notation of logic design. $$ \left(x_{1} \wedge \overline{x_{2}}\right) \vee\left(x_{1} \wedge x_{2}\right) \vee\left(\overline{x_{1}} \wedge x_{2}\right) $$

Short Answer

Expert verified
Use one OR gate to combine outputs of three AND gates with necessary NOT gates for negations.

Step by step solution

01

Identify the Terms

The given Boolean expression consists of three separate terms connected by the OR (鈭�) operator. These terms are: 1. \( x_1 \land \overline{x_2} \)2. \( x_1 \land x_2 \)3. \( \overline{x_1} \land x_2 \)
02

Recognize Each Term's Gate Equivalent

Translate each Boolean operation into logic gate representations:- The operation \(\land\) represents an AND gate.- The operation \(\lor\) represents an OR gate.- Negation (\(\overline{}\)) represents a NOT gate.
03

Begin Design with NOT Gates

Apply NOT gates to the input variables where necessary:- \( \overline{x_2} \): Input \( x_2 \) passes through a NOT gate.- \( \overline{x_1} \): Input \( x_1 \) passes through another NOT gate.
04

Connect AND Gates

Each pair of variables joined by \(\land\) needs an AND gate:- Connect \( x_1 \) and \( \overline{x_2} \) to an AND gate.- Connect \( x_1 \) and \( x_2 \) to another AND gate.- Connect \( \overline{x_1} \) and \( x_2 \) to a third AND gate.
05

Combine Outputs with OR Gates

The results of the AND gates are the inputs of the OR gates: - Connect the outputs of the three AND gates to a single OR gate. This OR gate's output represents the final expression in logic design notation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logic Gates
Logic gates are the building blocks of digital circuits. They manipulate binary signals, which exist in one of two states: high (1) or low (0). Here are the main types of logic gates:
  • AND Gate: This gate outputs a 1 only if all its inputs are 1. The logical expression is represented as a multiplication (e.g., \( x \land y \)).
  • OR Gate: This gate outputs a 1 if at least one of its inputs is a 1. The OR operation is denoted by \( x \lor y \).
  • NOT Gate: This gate inverts the input signal. Thus, a 0 becomes 1 and vice versa. The NOT operation is typically written over the variable, like \( \overline{x} \).
By understanding these basic gates, more complex functions and operations can be constructed in digital logic systems.
This foundation is crucial when diving into logic design using circuits.
Boolean Algebra
Boolean algebra forms the theoretical basis for digital circuits. It's similar to regular algebra, but it uses binary values and logical operations.
  • Operations: The main operations are AND, OR, and NOT. Their behavior can be summarized as:
    • AND (\( \land \)): Only true if both operands are true.
    • OR (\( \lor \)): True if at least one operand is true.
    • NOT (\( \overline{} \)): Converts true to false and vice versa.
  • Properties: Common properties of Boolean algebra include commutative, associative, and distributive properties, which help in simplifying expressions.
  • Simplification: Boolean expressions can often be simplified using these properties, which makes designing logical circuits more efficient.
Understanding Boolean algebra helps in translating logical statements into circuits and performing operations efficiently.
Logic Design
Logic design involves creating efficient digital circuits that implement specific Boolean expressions. This process uses logic gates to form a desired circuit output.
  • Translation Process: To design a circuit, each part of a Boolean expression is translated into corresponding gates:
    • AND operations become AND gates.
    • OR operations become OR gates.
    • NOT operations use NOT gates.
    Each gate represents a particular function of the logical expression.
  • Circuit Construction: The goal is to minimize the number of gates used, which requires skillful arrangement of the gates. Efficient logic design often involves simplifications gained from Boolean algebra.
  • Practical Application: Once designed, these circuits are implemented on hardware like Integrated Circuits (ICs), vital components in computers and various electronic devices.
Developing a strong understanding of logic design allows for the creation of highly efficient digital systems.

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Most popular questions from this chapter

Give an example of a Boolean algebra of order 16 whose elements are certain subsets of the set {1,2,3,4,5,6,7}

Prove if two finite sets \(A_{1}\) and \(A_{2}\) both have \(n\) elements then \(\left[\mathcal{P}\left(A_{1}\right) ; \cup, \cap, c\right]\) is isomorphic to \(\left[\mathcal{P}\left(A_{2}\right) ; \cup, \cap,^{c}\right]\)

Let \(\left[B ; V_{1} \wedge_{+}-\right]\) be a Boolean algebra of order \(4,\) and let \(f\) be a Boolean function of two variables on \(B\). (a) How many elements are there in the domain of \(\mathrm{f?}\) (b) How many different Boolean functions are there of two, variables? Three variables? (c) Determine the minterm normal form of \(f\left(x_{1}, x_{2}\right)=x_{1} \vee x_{2}\). (d) If \(B=\\{0, a, b, 1\\},\) define a function from \(B^{2}\) into \(B\) that is not a Boolean function.

We naturally order the numbers in \(A_{m}=\\{1,2, \ldots, m\\}\) with "less than or equal to," which is a partial ordering. We define an ordering, \(\preceq\) on the elements of \(A_{m} \times A_{n}\) by $$ (a, b) \preceq\left(a^{\prime}, b^{\prime}\right) \Leftrightarrow a \leq a^{\prime} \text { and } b \leq b^{\prime} $$ (a) Prove that \(\preceq\) is a partial ordering on \(A_{m} \times A_{n}\). (b) Draw the ordering diagrams for \(\preceq\) on \(A_{2} \times A_{2}, A_{2} \times A_{3},\) and \(A_{3} \times A_{3}\). (c) In general, how does one determine the least upper bound and greatest lower bound of two elements of \(A_{m} \times A_{n},(a, b)\) and \(\left(a^{\prime}, b^{\prime}\right) ?\) (d) Are there least and/or greatest elements in \(A_{m} \times A_{n} ?\)

State the dual of: (a) \(a \vee(b \wedge a)=a\). (b) \(a \vee(\overline{(\bar{b} \vee a) \wedge b})=1\). (c) \((\overline{a \wedge \bar{b}}) \wedge b=a \vee b\).
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