Boolean Algebra Chapter 5 Flashcards

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.

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.