# Two non-isomorphic structures with the same number of elements

[white paper: pedagogical]

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]

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]

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]

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]

# Goldbach Conjecture, Twin Primes Conjecture, and Bounded Gap Theorem in the language of number theory

[white paper: pedagogical]

We write the formulas of the theorem and the conjectures highlighted in the title of this white paper in the language of number theory for pedagogical purpose in first-order logic.

https://doi.org/10.31219/osf.io/u45y3

Open Mathematics Collaboration

Op. J. Math. Phys.
Volume 3, Article 229, 2021 [IZ]

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

[pedagogical]

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]

# Blended Teaching and Learning Methodology: Flipped Classroom, Peer and Problem-Based Instruction

[pedagogical]

This article presents a selection of ideas borrowed from active methodologiesâ€“namely, Flipped Classroom (FC), Peer Instruction (PI), and Problem Based Learning (PBL)â€“as a blended approach for teaching and learning physics and mathematics.

https://doi.org/10.31219/osf.io/f4qmr

Open Collaboration

Op. J. Math. Phys.
Volume 2, Article 71, 2020 [CS]

# Golden Rules for Referees

[pedagogical]

This manuscript serves as a pedagogical guideline for referees during the peer-reviewing process.

https://doi.org/10.31219/osf.io/grz9j

Open Collaboration

Op. J. Math. Phys.
Volume 1, Article 65, 2019 [CM]

# Teacher as a Researcher in Pure Mathematics

[pedagogical]

We propose, as an Idea Report, that a teacher well educated in research in pure mathematics can strengthen her/his teaching practice.

https://doi.org/10.31219/osf.io/849gh

Open Collaboration

Op. J. Math. Phys.
Volume 1, Article 61, 2019 [CI]

[pedagogical]