/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 Verify that each set of boolean ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 67

Verify that each set of boolean operators is functionally complete. $$\\{+, '\\}$$

Short Answer

Expert verified
The given set of Boolean operators {+, '} is functionally complete. We demonstrated that the AND operator can be expressed using the NOT (卢) and OR (+) operators with the expression \(A 鈭� B = 卢(卢A + 卢B)\). As the OR operator is already part of the given set, any Boolean function can be constructed using only these operators, proving functional completeness.

Step by step solution

01

Show that the AND operator can be expressed using the GIVEN operators.

We need to show that the AND operator can be represented using the NOT (卢) and OR (+) operators. The truth table for the AND operator (A 鈭� B) is as follows: | A | B | A 鈭� B | |---|---|-------| | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 0 | | 1 | 1 | 1 | We'll express the AND operator using the 卢 and + operators and compare it to the truth table of the AND operator. Our expression will have the general form (A # B), but use 卢 and + operators in place of the #. We can use the following expression: \(A 鈭� B = 卢(卢A + 卢B)\) Make sure to match any paranthesis with the respective boolean operator. Let's check the truth values for this expression: | A | B | 卢A | 卢B | 卢A + 卢B | 卢(卢A + 卢B) | |---|---|----|----|---------|-----------| | 0 | 0 | 1 | 1 | 1 | 0 | | 0 | 1 | 1 | 0 | 1 | 0 | | 1 | 0 | 0 | 1 | 1 | 0 | | 1 | 1 | 0 | 0 | 0 | 1 | As shown, the expression 卢(卢A + 卢B) has the same truth values as the AND operator. Thus, we have represented the AND operator using the 卢 and + operators.
02

Show that the other Boolean functions can be expressed using the GIVEN operators.

Since we've shown that the AND operator can be expressed using the 卢 and + operators, and the OR operator is already part of the given Boolean operator set, any Boolean function can be constructed using only these operators. This is because any Boolean function can be represented as a composition of AND and OR operators, alongside with NOT operators for negations. For instance, other Boolean functions, such as NAND (卢(A 鈭� B)) or XOR (A 鈭� 卢B + 卢A 鈭� B), can also be constructed using the given Boolean operators {+, '}. Therefore, the set of Boolean operators {+, '} is functionally complete.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Boolean Operators
Boolean operators are the building blocks of Boolean algebra, a subset of algebra used for creating true or false statements. These operators include AND, NOT, and OR, and they are used to combine or modify Boolean values and expressions. When working with digital circuits or computer programming, understanding Boolean operators is critical as they determine how different conditions interact with each other.

Crucial to the study of computer science and logic, these operators correspond to fundamental logic gates in circuit design and conditional structures in programming languages. The basic Boolean operators can perform all possible operations in Boolean algebra, demonstrating their functional completeness.
Truth Table
A truth table is a mathematical table used to compute the functional values of logical expressions on each combination of values taken by their logical variables. It showcases the output of every possible input combination, establishing the foundation for logic in computer science and mathematics.

Truth tables are essential for understanding the logical behavior of Boolean expressions and for designing logical circuits. They can be especially helpful to visualize how Boolean operators work and to prove equivalences or simplifications of Boolean expressions, as demonstrated in verifying functional completeness.
Logical Conjunction (AND)
Logical conjunction, denoted by the symbol AND or \(\land\), is a fundamental Boolean operator that results in true if both operands are true, and false otherwise. In simple terms, if you think of each operand as a condition, an AND operation only yields a true result when both conditions are met.

Considering the common usage of AND in everyday language, such as in statements that require multiple criteria to be true, its application in Boolean logic is quite intuitive. In digital electronics, an AND gate would output a high signal only when all of its inputs are high.
Logical Negation (NOT)
Logical negation is symbolized by the word NOT or \(eg\), and it represents the inversion operation in Boolean algebra. It flips the truth value of the operand鈥攖urning true into false, and vice versa. The negation operator is unary, meaning it operates on a single operand.

For instance, if a particular statement is false, NOT applied to it would make it true. NOT operations are crucial for creating complementary conditions and are represented by inverters in digital circuits.
Logical Disjunction (OR)
Logical disjunction is signified by the OR operator or \(\lor\), and it results in true if at least one of the operands is true, allowing it to be used for situations where any of multiple conditions being true warrants the same outcome.

This operator is associative, which means that the result remains the same regardless of how the operands are grouped. In digital circuitry, an OR gate outputs a high signal if any of its inputs are high. It鈥檚 often used in logic and decision-making processes where multiple possible conditions lead to the same conclusion.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the DNF of each boolean function. $$ f(x, y, z)=y(x+z) $$

Simplify each boolean expression using the laws of boolean algebra. $$w x y z+w^{\prime} x y^{\prime} z^{\prime}+w x y z^{\prime}+w^{\prime} x y^{\prime} z$$

Find the DNF of each boolean function. $$ f(x, y, z)=x+y z^{\prime} $$

Using a Karnaugh map, simplify each boolean expression. $$w x y z+w x y z^{\prime}+w x y^{\prime} z^{\prime}+w x y^{\prime} z+w x^{\prime} y^{\prime} z+w^{\prime} x^{\prime} y^{\prime} z+w^{\prime} x y^{\prime} z$$

Evaluate each boolean expression. $$(1 \downarrow 0) \uparrow(1 \downarrow 1)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.