Boolean Algebra Chapter 3 Flashcards

Questions and Answers

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?

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?

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?

Boolean addition is denoted by + and is equivalent to the logical ∨ ('or') operation. Signup and view all the answers

What is the notation for Boolean complement and what logical equivalent statement does it have?

The complement is denoted with a bar symbol and is equivalent to the ¬ ('not') operation. Signup and view all the answers

What is the notation for Boolean exclusive or and what logical equivalent statement does it have?

The exclusive or (XOR) is denoted by , and is equivalent to p XOR q = (-p ^ q) V (p ^ -q). Signup and view all the answers

What are the idempotent laws in Boolean Algebra?

x + x = x and x · x = x. Signup and view all the answers

What are the associative laws in Boolean Algebra?

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

What are the commutative laws in Boolean Algebra?

x + y = y + x and xy = yx. Signup and view all the answers

What are the distributive laws in Boolean Algebra?

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

What are the identity laws in Boolean Algebra?

x + 0 = x and x · 1 = x. Signup and view all the answers

What are the domination laws in Boolean Algebra?

x · 0 = 0 and x + 1 = 1. Signup and view all the answers

What is the double complement law in Boolean Algebra?

--x = x. Signup and view all the answers

What are the complement laws in Boolean Algebra?

x · x' = 0 and x + x' = 1. Signup and view all the answers

What are De Morgan's laws?

-(x + y) = -x · -y and -(xy) = -x + -y. Signup and view all the answers

What are the absorption laws in Boolean Algebra?

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

What are the three methods that exist to prove that two Boolean expressions are equivalent?

Deduction, Truth Table, Venn diagram. Signup and view all the answers

A _______ function maps one or more Boolean input values to the set {0, 1}.

Boolean Signup and view all the answers

What are the two forms of Boolean expressions?

Disjunctive Normal Form and Conjunctive Normal Form. Signup and view all the answers

What is the definition of Disjunctive Normal Form (DNF)?

A DNF expression has the form c1 + c2 +....+ cm, where each cj is a product of literals. Signup and view all the answers

What are the rules of Disjunctive Normal Form?

The complement can only be applied to a single variable, and no addition within terms. Signup and view all the answers

What are the rules of Conjunctive Normal Form?

A CNF expression has the form d1 · d2 · ... · dm where each dj is a sum of literals; complement is only applied to single variables and no multiplication within a factor. Signup and view all the answers

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.

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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.