Boolean Algebra Chapter 5 Flashcards

Questions and Answers

What is Boolean algebra?

A set of rules for numerical operations

A mathematical structure dealing with real numbers

A set of rules and operations for working with variables whose values are either 0 or 1 (correct)

A programming language for logic circuits

What value does 0 correspond to in propositional logic?

What value does 1 correspond to in propositional logic?

What does Boolean multiplication represent?

Logical AND operation Signup and view all the answers

What operation does Boolean addition represent?

Logical OR operation Signup and view all the answers

What does the complement of an element do?

Reverses that element's value Signup and view all the answers

What are Boolean variables?

Variables that can have a value of 1 or 0 Signup and view all the answers

What are Boolean expressions formed from?

Boolean variables and operations Signup and view all the answers

What does it mean for two Boolean expressions to be equivalent?

They have the same value for every combination of variable values Signup and view all the answers

Which of the following represents the Idempotent law in Boolean algebra?

x + x = x and x * x = x Signup and view all the answers

What is the Associative law in Boolean algebra?

(x + y) + z = x + (y + z) and (xy)z = x(yz) Signup and view all the answers

What does the Distributive law state?

x * (y + z) = xy + xz Signup and view all the answers

What does the Double complement law state?

x(double complement) = x Signup and view all the answers

What are De Morgan's laws?

(x + y)(complement) = x(complement) * y(complement) and (xy)(complement) = x(complement) + y(complement) Signup and view all the answers

What is a Boolean function?

A function mapping Boolean input values to the set {0, 1} Signup and view all the answers

What does an input/output table represent?

A truth table showing the output for all possible combinations of inputs Signup and view all the answers

What is a literal?

A Boolean variable or its complement Signup and view all the answers

What is a minterm in Boolean functions?

A product of literals associated with specific input values Signup and view all the answers

What is disjunctive normal form (DNF)?

A Boolean expression in the sum of products form Signup and view all the answers

What is conjunctive normal form (CNF)?

A Boolean expression in the product of sums form Signup and view all the answers

Study Notes

Boolean Algebra Overview

Boolean algebra comprises rules and operations for manipulating variables that hold binary values: 0 and 1.
Boolean variables only take on values of 0 or 1.

Values in Boolean Algebra

The value 0 represents false (F) in propositional logic.
The value 1 represents true (T) in propositional logic.

Boolean Operations

Boolean Multiplication (AND): Denoted by a dot or adjacency; behaves like standard multiplication but results correspond to the logical AND operation.
Boolean Addition (OR): Denoted by a plus sign (+); follows standard addition rules with the exception that 1 + 1 results in 1, corresponding to logical OR.

Complement and Variables

The complement of an element reverses its value, denoted with a bar or apostrophe (x’), similar to the logical NOT operation.
Boolean expressions combine Boolean variables and constants (0 and 1) using established operations with a specific order of precedence.

Equivalence and Properties

Two Boolean expressions are equivalent if they yield identical outcomes across all variable combinations, denoted by the equal sign (=).
Idempotent Laws: x + x = x and x * x = x.
Associative Laws: (x + y) + z = x + (y + z) and (xy)z = x(yz).
Commutative Laws: x + y = y + x and xy = yx.
Distributive Laws: x + yz = (x + y)(x + z) and x(y + z) = xy + xz.

Identity, Domination, and Complement Laws

Identity Laws: x + 0 = x and x * 1 = x.
Domination Laws: x * 0 = 0 and x + 1 = 1.
Double Complement Law: The double complement of x yields x.
Complement Laws: x * x' = 0 and x + x' = 1.

De Morgan's Laws

(x + y)’ = x’ * y’ and (xy)’ = x’ + y’.

Absorption Laws

x + (xy) = x and x(x + y) = x.

Boolean Functions

A Boolean function maps input values (from {0, 1}) to outputs in the same set.
Represented using input/output tables or Boolean expressions; for example, f(x, y, z) = xy + yz.

Additional Concepts

Input/Output Table: A truth table showing output values for each combination of inputs, typically started with 0s.
Literal: A Boolean variable or its complement.
Minterm: A specific product of literals resulting in unique combinations of variables.
Disjunctive Normal Form (DNF): A sum of products format for Boolean expressions (e.g., x'yz' + xy + w' + yz'w).
Conjunctive Normal Form (CNF): A product of sums format for Boolean expressions.

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Description

Explore the essential concepts of Boolean algebra with these flashcards. Each card provides a definition for key terms like Boolean algebra, values 0 and 1, and Boolean multiplication. Perfect for grasping the foundational elements of this vital topic in mathematical logic.