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Questions and Answers
What are the rules of precedence of Boolean Algebra?
What are the rules of precedence of Boolean Algebra?
Boolean multiplication takes precedence over Boolean addition. The complement operation is applied as soon as the entire expression under the bar is evaluated. Parentheses can be used to override the precedence rules.
What is the definition of Boolean Algebra?
What is the definition of Boolean Algebra?
A set of rules and operations for working with variables whose values are either 0 or 1.
What is the notation for Boolean multiplication and what logical equivalent statement does it have?
What is the notation for Boolean multiplication and what logical equivalent statement does it have?
Boolean multiplication is denoted by and is equivalent to the logical ∧ ('and') operation.
What is the notation for Boolean addition and what logical equivalent statement does it have?
What is the notation for Boolean addition and what logical equivalent statement does it have?
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What is the notation for Boolean complement and what logical equivalent statement does it have?
What is the notation for Boolean complement and what logical equivalent statement does it have?
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What is the notation for Boolean exclusive or and what logical equivalent statement does it have?
What is the notation for Boolean exclusive or and what logical equivalent statement does it have?
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What are the idempotent laws in Boolean Algebra?
What are the idempotent laws in Boolean Algebra?
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What are the associative laws in Boolean Algebra?
What are the associative laws in Boolean Algebra?
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What are the commutative laws in Boolean Algebra?
What are the commutative laws in Boolean Algebra?
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What are the distributive laws in Boolean Algebra?
What are the distributive laws in Boolean Algebra?
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What are the identity laws in Boolean Algebra?
What are the identity laws in Boolean Algebra?
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What are the domination laws in Boolean Algebra?
What are the domination laws in Boolean Algebra?
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What is the double complement law in Boolean Algebra?
What is the double complement law in Boolean Algebra?
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What are the complement laws in Boolean Algebra?
What are the complement laws in Boolean Algebra?
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What are De Morgan's laws?
What are De Morgan's laws?
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What are the absorption laws in Boolean Algebra?
What are the absorption laws in Boolean Algebra?
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What are the three methods that exist to prove that two Boolean expressions are equivalent?
What are the three methods that exist to prove that two Boolean expressions are equivalent?
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A _______ function maps one or more Boolean input values to the set {0, 1}.
A _______ function maps one or more Boolean input values to the set {0, 1}.
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What are the two forms of Boolean expressions?
What are the two forms of Boolean expressions?
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What is the definition of Disjunctive Normal Form (DNF)?
What is the definition of Disjunctive Normal Form (DNF)?
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What are the rules of Disjunctive Normal Form?
What are the rules of Disjunctive Normal Form?
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What are the rules of Conjunctive Normal Form?
What are the rules of Conjunctive Normal Form?
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Study Notes
Boolean Algebra Overview
- Boolean Algebra is a mathematical structure for operations on binary values (0 and 1).
- Utilizes logical operators including AND (∧), OR (∨), and NOT (¬).
Rules of Precedence
- Boolean multiplication has higher precedence than Boolean addition.
- The complement operation is applied immediately after evaluating the expression under the bar.
- Parentheses can modify precedence rules.
Boolean Operations
- Boolean Multiplication: Denoted by juxtaposition, equivalent to the logical AND operation (p ∧ q).
- Boolean Addition: Denoted by +, equivalent to logical OR (p ∨ q), except that 1 + 1 does not equal 2.
- Boolean Complement: Denoted by a bar (e.g., x̅), inverses the value (0 becomes 1 and vice versa), functions analogously to NOT.
- Boolean Exclusive OR (XOR): Outputs 1 when inputs differ, denoted by ⊕, related to p XOR q = (¬p ∧ q) ∨ (p ∧ ¬q).
Fundamental Laws of Boolean Algebra
-
Idempotent Laws:
- x + x = x
- xx = x
-
Associative Laws:
- (x + y) + z = x + (y + z)
- (xy)z = x(yz)
-
Commutative Laws:
- x + y = y + x
- xy = yx
-
Distributive Laws:
- x + yz = (x + y)(x + z)
- x(y + z) = xy + xz
-
Identity Laws:
- x + 0 = x
- x1 = x
-
Domination Laws:
- x0 = 0
- x + 1 = 1
-
Double Complement Law:
- x̅̅ = x
-
Complement Laws:
- xx̅ = 0
- 1x̅ = 0
- x + x̅ = 1
- 0 + 1 = 1
-
De Morgan's Laws:
- ¬(x + y) = ¬x · ¬y
- ¬(xy) = ¬x + ¬y
-
Absorption Laws:
- x + (xy) = x
- x(x + y) = x
Equivalence of Boolean Expressions
-
Methods to Prove Equivalence:
- Deduction: Applying Boolean laws to show equality.
- Truth Table: Highlighting identical truth values across expressions.
- Venn Diagram: Illustrating equivalence through overlapping sets.
Boolean Functions
- A Boolean function translates one or more binary input values into the set {0, 1}.
Forms of Boolean Expressions
- Disjunctive Normal Form (DNF): Structured as a sum of products (c1 + c2 + … + cm), where each term c is a product of literals.
- Conjunctive Normal Form (CNF): Formulated as a product of sums (d1 · d2 · … · dm), with each term d as a sum of literals.
Rules for Normal Forms
-
Disjunctive Normal Form (DNF) Rules:
- Only apply complements to a single variable.
- Avoid addition within individual terms.
-
Conjunctive Normal Form (CNF) Rules:
- Complement applied to single variables.
- Prohibit multiplication within factors.
Studying That Suits You
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Description
Test your knowledge of Boolean Algebra with these flashcards based on Chapter 3. Learn key concepts like rules of precedence, operations, and definitions relevant to this foundational topic. Perfect for students seeking to reinforce their understanding of Boolean logic.