Question

Write the duals of the following Boolean equations: (1) \((a+b)+c(1+d)+1=1\) (2) \(\mathrm{D} \cdot\left(\underline{\mathrm{A}}+\mathrm{B}^{-}\left(\underline{ \left.\mathrm{C}+\mathrm{D}^{-}\right)}\right)=\underline{\mathrm{A}} \mathrm{D}(\mathrm{B}+\underline{\mathrm{C}})\right.\)

Short answer

The duals of the given Boolean equations are: (1) \((a\cdot b)+c(0\cdot d)+0=0\) (2) \(D+\overline{A}B^{+1}+\overline{C}D^{+1}=\overline{A}+D(B\cdot\overline{C})\)

Step-by-step solution

Identify the operations to exchange

In the first expression, we have the operations '+' (OR) and '()' (AND), as well as the constants '1' and '0'. The second expression contains '⋅' (AND), '+' (OR), '⏞' (AND), and underlined variables, which are complements of variables using the bar notation. We must replace OR operations with AND operations and vice versa, swap '1's and '0's, and exchange products and sums. #Step 2: Find the dual of the first Boolean equation#

Find the dual of the first Boolean equation

Given equation: \((a+b)+c(1+d)+1=1\) To find its dual, replace: - '+' with '⋅' - '()' with '+' - '1' with '0' \[ (a \cdot b)+(c+0 \cdot d)+0=0 \] So, the dual of the first Boolean equation is: \((a\cdot b)+c(0\cdot d)+0=0\) #Step 3: Find the dual of the second Boolean equation#

Find the dual of the second Boolean equation

Given equation: \(D \cdot(\underline{A}+ B^{-}(\underline{C} + D^{-}) )=\underline{A}D(B +\underline{C})\) To find its dual, replace: - '⋅' with '+' - '+' with '⋅' - '⏞' with '+' - Underlined variables with their complements (remove the underline and change the power of -1) \[D+(\overline{A}\cdot B^{-1}+\overline{C}\cdot D^{-1})=\overline{A}+D\cdot(B\cdot C^{-1})\] So, the dual of the second Boolean equation is: \(D+\overline{A}B^{+1}+\overline{C}D^{+1}=\overline{A}+D(B\cdot\overline{C})\) In conclusion, the duals of the given Boolean equations are: (1) \((a\cdot b)+c(0\cdot d)+0=0\) (2) \(D+\overline{A}B^{+1}+\overline{C}D^{+1}=\overline{A}+D(B\cdot\overline{C})\)

Key concepts

Boolean Equations

Boolean algebra is a branch of algebra centered around the manipulation of logical values or states, typically true (1) and false (0). Boolean equations are expressions that equate two Boolean expressions, using logical operations such as AND, OR, and NOT. These equations play a crucial role in computer science and electrical engineering, particularly in designing digital circuits and systems.

To solve a Boolean equation, we apply a series of rules and laws that are unique to Boolean algebra. Unlike traditional algebra, we work only with binary values where operations have different implications. Key among these operations are the logical AND (multiplication, denoted by '⋅'), OR (addition, denoted by '+'), and NOT (denoted by an overbar or underline for the complement of a variable). Identifying and manipulating these operations correctly is essential for solving and simplifying Boolean equations.

Understanding the duality principle in Boolean algebra can greatly enhance a student's ability to tackle these problems. The duality principle states that every Boolean equation remains valid if we swap OR and AND operations, as well as 1's and 0's. Therefore, finding the dual of a given Boolean equation involves systematically exchanging these elements.

De Morgan's Laws

One of the foundational concepts in Boolean algebra are De Morgan's laws. These laws provide a systematic way of translating an expression with logical ANDs into one with logical ORs, and vice versa, by also negating the involved variables. Specifically, De Morgan's laws state that the complement of the AND of two variables is equivalent to the OR of the complements of those variables, and the complement of the OR of two variables is equivalent to the AND of the complements of those variables.

Written in formal logic notation, De Morgan's laws are:

• The complement of (A AND B) is equal to (NOT A) OR (NOT B): \( \overline{A \cdot B} = \overline{A} + \overline{B} \)
• The complement of (A OR B) is equal to (NOT A) AND (NOT B): \( \overline{A + B} = \overline{A} \cdot \overline{B} \)

Applying De Morgan's laws is crucial when simplifying Boolean expressions, particularly when one must negate a complex logical statement. They also play an important role in the synthesis and analysis of digital circuits, where the physical implementation of NOT operations can be costly or unwieldy.

Logic Operations

Logic operations are the building blocks of Boolean algebra and determine the outcome of Boolean expressions based on the values of their variables. The three primary logic operations are AND, OR, and NOT.

Here's a brief overview of each:

• AND (⋅): This operation yields true only if both operands are true. In digital electronics, it's represented by a series connection.
• OR (+): True if either or both operands are true, represented by a parallel connection.
• NOT: A unary operation that inverts the value of its operand. True becomes false, and vice versa.

Each of these operations has a corresponding logical gate in hardware implementations. By combining these gates, complex digital circuits can be created to perform a variety of functions, from simple switches to components of a modern microprocessor.

In Boolean equations, the use of these operations must respect specific laws and properties, such as commutative, associative, distributive, and absorption laws, to correctly manipulate the equations to achieve the simplest or most desired form. Understanding how to apply these operations within the context of an equation can be crucial for success in various applications, such as creating efficient software algorithms or optimizing circuit designs.