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Showing 31–60 of 1709 reference entries

Physics · MeasurementCaesium-133 defining frequency

ΔνCs denotes the unperturbed ground-state hyperfine transition frequency of caesium-133. Its numerical value is fixed exactly in SI units.

DefinitionMathematics & logic · InformationClassical propositions and connectives

Classical logic combines truth-valued propositions using defined connectives.

DefinitionMathematics & logicAxioms, deduction and mathematical proof

A mathematical proof derives a conclusion within stated assumptions and rules.

MethodMathematics & logic · InformationSets, membership and basic operations

A set specifies membership without counting duplicate listings.

DefinitionMathematics & logic · InformationFunctions, domain and codomain

A function assigns exactly one output to each input in its domain.

DefinitionMathematics & logic · InformationMathematical induction

A base case and general successor step establish a predicate on all nonnegative integers.

MethodMathematics & logic · MeasurementEuclidean right triangles and the Pythagorean theorem

In a Euclidean right triangle, squared leg lengths sum to the squared hypotenuse length.

Mathematical statementMathematics & logic · PhysicsSine, cosine and the unit circle

Sine and cosine are unit-circle coordinates whose squared values sum to one.

Mathematical statementMathematics & logic · InformationComplex numbers

A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.

DefinitionMathematics & logic · Information · PhysicsLinear maps and matrix representation

A linear map preserves linear combinations and has a matrix once bases are chosen.

DefinitionMathematics & logicLimit of a real function

A finite limit controls output error through sufficiently small input distance.

DefinitionMathematics & logic · PhysicsDerivative at a point

A derivative is a finite limit of local difference quotients.

DefinitionMathematics & logic · PhysicsDefinite integral and signed accumulation

A definite integral is the limiting signed accumulation of small contributions.

DefinitionMathematics & logic · PhysicsFundamental theorem of calculus

Continuity connects differentiation with integration.

Mathematical statementMathematics & logic · InformationProbability space and axioms

A probability assigns nonnegative, normalised, countably additive weights to events.

DefinitionMathematics & logic · InformationConditional probability and independence

Conditioning restricts attention to an event of positive probability.

DefinitionMathematics & logic · InformationExpected value of a discrete random variable

Expected value weights possible values by their probabilities.

DefinitionMathematics & logic · MeasurementConfidence interval for a population mean

A confidence level describes repeated-sampling coverage of an interval procedure.

MethodInformation · Mathematics & logicBits and unsigned binary encoding

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

DefinitionInformation · Mathematics & logicShannon entropy of a discrete distribution

Shannon entropy measures uncertainty in a specified probability distribution.

DefinitionInformationAlgorithm and decision procedure

An algorithm specifies effective steps; a decision procedure must terminate on every permitted input.

MethodInformation · Mathematics & logicUndecidability of the halting problem

No total algorithm decides halting for every program and input in a general computation model.

Mathematical statementInformation · Mathematics & logicAsymptotic upper bounds and big O

Big O bounds eventual growth up to a fixed constant factor.

DefinitionMeasurementSI base units

Seven named base units organize the SI quantity system.

DefinitionMeasurementSI decimal prefixes

Prefix symbols specify decimal scale factors.

DefinitionMeasurementMetrological traceability

Connect a result to a reference through documented calibrations.

DefinitionMeasurement · Mathematics & logicFirst-order uncertainty propagation

Combine input uncertainty through a measurement model.

MethodMeasurement · Physics · Mathematics & logicDimensional consistency

Use dimensions to detect impossible quantity equations.

MethodPhysics · Earth & cosmosNewtonian gravitation

An inverse-square force models attraction between point masses.

ModelPhysicsFrequency, wavelength and phase speed

Relate a periodic wave's spatial and temporal repetition.

Definition
DefinitionSource checked

Classical propositions and connectives

Mathematics & logicInformation

Classical logic combines truth-valued propositions using defined connectives.

Key knowledge

In classical two-valued logic a proposition is true or false. Negation reverses its truth value; a conjunction is true when both parts are true, an inclusive disjunction when at least one is true, and an implication is false only when its antecedent is true and consequent false.

Conditions and scope

  • These are classical truth-functional meanings; other logical systems use different semantics.
  • Logical implication alone does not assert physical causation.
Sources · 3

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Contemporary Mathematics, §2.1: StatementsOpenStax, Rice UniversityTextbook chapteropenstax.orgRead source
  2. Contemporary Mathematics, 2.2: Compound statementsOpenStaxTextbook chapteropenstax.orgRead source
  3. Discrete Mathematics II · Set theory for computer scienceGlynn Winskel, University of CambridgeTeaching resource · PDFcl.cam.ac.ukRead source

Evidence and review

Two-valued proposition and connective definitions.

2026-10-10 · Checked truth conditions and inclusive-or convention.

Foundational review

2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.

Sources from other places

Agreement within this scope

Classical negation, conjunction, inclusive disjunction and material implication.

The inspected truth table agrees; other logical semantics are outside this comparison.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–9 of 9 relationships

Scientific relationships · 9 / 9

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • A claim that P(x) holds for every x in a specified domain is false if an element of that domain makes P(x) false. Testing some successful examples does not establish the claim for an infinite domain.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • Axioms are assumptions accepted within a mathematical system. A proof derives a proposition from those assumptions and previously established results through permitted logical deductions.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • In classical first-order logic, ∀x φ holds when φ holds for every domain element; ∃x φ holds when it holds for at least one. Free variables require an assignment.
  4. Uses definition
    Relationship sources · 1

    Linked statements

    • For a 3CNF formula with m clauses, make a vertex for each literal occurrence. Join occurrences from different clauses unless their literals are complementary. The formula is satisfiable exactly when the graph has a clique of size m: choose one true occurrence per clause; conversely the clique's consistent literals extend to an assignment.
  5. Uses definition
    Relationship sources · 1

    Linked statements

    • A DFA's language is empty iff no accepting state is reachable from its start. Two DFAs are equivalent iff their product automaton's symmetric-difference language is empty. Product-state accepting rules also construct regular intersections and unions; swapping accepting/nonaccepting states complements a complete DFA.
  6. Uses definition
    Relationship sources · 2

    Linked statements

    • NL=coNL: complements of nondeterministic logspace decision languages also lie in NL. The reachability proof certifies counts of reachable vertices, making nonreachability testable with nondeterministic logarithmic workspace.
  7. Uses definition
    Relationship sources · 1

    Linked statements

    • Boolean satisfiability SAT is NP-complete under polynomial-time many-one reductions. For each language A∈NP and input x, a polynomial-size formula encodes a polynomial-time accepting computation tableau so that x∈A exactly when that formula is satisfiable; a satisfying assignment itself supplies the NP witness.
  8. Uses definition
    Relationship sources · 1

    Linked statements

    • Introduce variables for Boolean subformula gates, constrain each gate to agree with its inputs using clauses of at most three literals, and require the output gate true. This constructs a linear-size equisatisfiable 3CNF formula, showing SAT≤ₚ3SAT and therefore 3SAT is NP-complete.
  9. Related to
    Relationship sources · 1

    Linked statements

    • AND returns 1 exactly when both inputs are 1; inclusive OR returns 1 when at least one is 1; NOT exchanges 0 and 1. Two-input XOR returns 1 exactly when inputs differ. A truth table specifies every input combination; AND, OR and NOT suffice to express any finite-input Boolean function.

    Boolean gate AND, inclusive OR and NOT implement conjunction, disjunction and negation on truth values.

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