Questions and Answers
What is Boolean algebra?
What value does 0 correspond to in propositional logic?
F
What value does 1 correspond to in propositional logic?
T
What does Boolean multiplication represent?
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What operation does Boolean addition represent?
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What does the complement of an element do?
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What are Boolean variables?
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What are Boolean expressions formed from?
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What does it mean for two Boolean expressions to be equivalent?
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Which of the following represents the Idempotent law in Boolean algebra?
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What is the Associative law in Boolean algebra?
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What does the Distributive law state?
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What does the Double complement law state?
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What are De Morgan's laws?
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What is a Boolean function?
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What does an input/output table represent?
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What is a literal?
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What is a minterm in Boolean functions?
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What is disjunctive normal form (DNF)?
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What is conjunctive normal form (CNF)?
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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.