Personal Handbook of Logic

This is a personal collection of definitions and results from first-order logic.

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 252, 2021 [JX]

Categories in symbols

An introduction to the language of category theory is presented in a very minimalistic fashion, using the fewest number of symbols as possible. In this white paper, the focus is on the main concepts underlying categories.

https://doi.org/10.31219/osf.io/k9fbc

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 247, 2021 [JR]

Lyapunov exponents for the Lorentz transformations

[microresearch]

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We calculate the Lyapunov exponents of the complex stretch factor $f^-(z)=(1-z^2)^{-1/2}$ from the Lorentz transformations and of its reciprocal $f^+(z)=(1-z^2)^{+1/2}$ as well.

https://doi.org/10.31219/osf.io/jsncm

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 239, 2021 [JJ]

Time is a discrete dynamical system

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We conjecture that quantum vacuum operates its discrete dynamics in a superposition of a class of iterating functions such that each physical system operates within a distinct function.

https://doi.org/10.31219/osf.io/8f4yg

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 238, 2021 [JI]

Two non-isomorphic structures with the same number of elements

[white paper: pedagogical]

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For pedagogical purposes, we define a simple language to show that two different structures with the same number of elements in their universes are not isomorphic.

https://doi.org/10.31219/osf.io/ytcru

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 234, 2021 [JE]

The isomorphism between structures is an equivalence relation

[white paper: pedagogical]

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We present a pedagogical proof that the function of an isomorphism between two structures is an equivalence relation.

https://doi.org/10.31219/osf.io/7uwnh

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 233, 2021 [JD]

The terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols

[white paper: pedagogical]

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We show using induction on complexity that all terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols.

https://doi.org/10.31219/osf.io/ue32a

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 231, 2021 [JB]

The power set has 2^n elements

[white paper: pedagogical]

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We prove that if A is a set consisting of n elements, then A has 2^n subsets.

https://doi.org/10.31219/osf.io/mptx7

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 230, 2021 [JA]

A linear algebra

[knowledge base]

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LINEAR ALGEBRA and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/8emp4

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 225, 2021 [IV]

Linear Transformations

[knowledge base]

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LINEAR TRANSFORMATION and its underlying definitions are presented in this white paper [knowledge base (http://omkb.org)].

doi.org/10.31219/osf.io/cjdwg

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 224, 2021 [IU]

Every group is isomorphic to a group of permutations

[knowledge base]

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CAYLEY’s THEOREM, the SYMMETRIC GROUP THEOREM, and their underlying definitions are presented in this white paper (knowledge base = http://omkb.org).

https://doi.org/10.31219/osf.io/63pmy

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 223, 2021 [IT]

A semigroup with a left identity and left inverse is a group

[pedagogical]

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We translate the proof of the theorem stated in the title, accomplished by Prover9, into a human readable form.

https://doi.org/10.31219/osf.io/wsbne

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 222, 2021 [IS]

The membership relation

[knowledge base]

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The MEMBERSHIP RELATION and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/r9p2m

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 221, 2021 [IR]

Modules over Rings

[knowledge base]

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MODULES over RINGS and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/xjsc7

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 220, 2021 [IQ]

Vector space over a field

[knowledge base]

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VECTOR SPACE OVER A FIELD and its underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/g6ux5

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 219, 2021 [IP]

Field, commutative ring, integral domain

[knowledge base]

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FIELD, three propositions, and their underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/wmjst

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 217, 2021 [IN]

RINGS: Almost a ring, semiring, zero, integral domain

[knowledge base]

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RING, commutative ring, almost a ring, semiring, zero ring, zero property, zero divisors, domain, integral domain, and their underlying definitions are presented in this white paper (knowledge base).

https://doi.org/10.31219/osf.io/bzugr

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 216, 2021 [IM]