Mathematical formulas

Algebra

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1Numbers and algebraic properties
2Powers, exponents, and radicals
3Algebraic expressions
4Notable products and identities
5Factorization
6Rational expressions and partial fractions
7Equations
8Inequalities
9Systems of equations
10Functions
11Polynomials
12Exponentials and logarithms
13Complex numbers
14Sequences, finite series, and recurrences
15Vectors
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17Determinants and inverses
18Vector spaces
19Linear transformations
20Eigenvalues and diagonalization
21Orthogonality and projections
22Least squares
23Matrix decompositions
24Matrix norms and conditioning
25Boolean algebra
26Modular arithmetic
27Algebraic structures
28Linear codes and error correction
29Relationship maps

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Subjects
Algebra
Boolean algebra

Section 25

Boolean algebra

Boolean identities

fundamentalviz

A + 0 = A; A 1 = A; A + 1 = 1; A 0 = 0

Basic identities for OR and AND.

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Idempotence and complement

fundamentalviz

A + A = A; AA = A; A + \overset{ˉ}{A} = 1; A \overset{ˉ}{A} = 0

Essential rules for Boolean simplification.

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Boolean distributivity

fundamentalviz

A (B + C) = A B + A C; A + B C = (A + B) (A + C)

Distributive algebra for logic.

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De Morgan's laws

fundamentalviz

\overline{A + B} = \overset{ˉ}{A} \overset{ˉ}{B}; \overline{A B} = \overset{ˉ}{A} + \overset{ˉ}{B}

Transforms negations of OR and AND.

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fundamentalviz

A \oplus B = \overset{ˉ}{A} B + A \overset{ˉ}{B}

It is true when exactly one input is true.

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fundamentalviz

A + A B = A; A (A + B) = A

Simplifies redundant expressions.

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Sum of products

intermediateviz

F = Σ m (i_{1}, \dots, i_{k})

Canonical form as an OR of minterms.

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Product of sums

intermediateviz

F = Π M (i_{1}, \dots, i_{k})

Canonical form as an AND of maxterms.

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Boolean equivalence

intermediateviz

F \equiv G ⟺ F y G tienen la misma tabla de verdad

Criterion for checking logical simplifications.

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