Solution. We prove that n + m is a natural number using mathematical induction on n. • (Base case) When n = 0, we have 0 + m = m ∈ N for all m ∈ N by definition of addition. • (Induction step) Suppose that n + m ∈ N for all m ∈ N. What we want to prove is that (n++) + m ∈ N for all m ∈ N. But since n + m ∈ N by the induction hypothesis, (PA2) shows that (n++) + m = (n + m)++ ∈ N as desired. Thi...

The article gives a definition of the concept “plot-didactic game”, characterizes such games, taking into account their use in process forming elementary mathematical representations, provides brief description games aimed at mastering program material section “Formation representations”. In addition, role acquisition concepts by preschool children is revealed.

Humans recognize themselves as a part of humanity, of the biological world and of life simultaneously. They identify themselves differently in each case. However, in the absence of an accurate definition of life, the process of the identification of humanity remains incomplete. The incompleteness of this process justifies the numerous attempts to define life. Nevertheless, none of these previou...

Computing the minimum distance of a lattice (and vectors achieving the minimum distance) is a fundamental problem in the algorithmic study of lattices, and the core of many applications in computer science and cryptography. Today, we will focus on mathematical properties of lattices, and answer questions like: is the quantity λ(Λ) always achieved by a pair of vectors? (I.e., can the inf in the ...

The question of what should be meant by a measurement is tackled from mathematical perspective whose physical interpretation that fundamental process via which finite amount classical information produced. This translates into an algebraic and topological definition space caters for the distinction between quantum measurements allows notion observer to derived.

a new definition of boundedness of linear order-homomorphisms (loh)in $l$-topological vector spaces is proposed. the new definition iscompared with the previous one given by fang [the continuity offuzzy linear order-homomorphism, j. fuzzy math. 5 (4) (1997)829$-$838]. in addition, the relationship between boundedness andcontinuity of lohs is discussed. finally, a new uniform boundednessprincipl...

We present and discuss a new axiom, Martin’s Maximum∗,++, cf. Definition 2.10, which amalgamates Woodin’s Pmax axiom (∗) and Martin’s Maximum++. If a mathematical object can be imagined in a reasonable way, then it exists!

In this note, we outline a definition of propositional manifold and logical cohomology. An application is also considered for mathematics, two Boole algebras of mathematical propositions are non equivalent if their two cohomologies are not isomorphic.