Recent content by enigmahunter

The "http://en.wikipedia.org/wiki/Continuum_hypothesis" " is my choice. "There is no set whose size is strictly between that of the integers and that of the real numbers." It is an interesting problem and considered as a fact, even though it cannot be proved or disproved in ZFC.

An injective map implies a well-defined map, but a well-defined map does not necessarily imply an injective map. f:X \longrightarrow Y a,b \in X and f(a), f(b) \in Y For a well-defined map, a=b implies f(a)=f(b). (if "a=b implies f(a) \neq f(b)", then f is not a function ). For...

NP is just the derived concept from a nondeterministic Turing machine. The interesting problems of NP are the NP complete problems. Every problems in NP are polynomial time reducible to the problems in this completeness class. If we find a polynomial time algorithm for any of NP complete...

First, I assume you are talking about permutations in a symmetric group. Fact 1. Every permutation in Sn can be written as a product of transpositions. Fact 2. A permutation in Sn (n>=2) cannot be both even and odd (number of transpositions). Fact 3. The set of all even permutations of Sn...

A model of the http://en.wikipedia.org/wiki/Peano_axioms" for natural numbers is a triple (N,0,S), where N is an infinite set and 0 belongs to N, S is a successor function. S(a) = a\cup\{a\} 0=\varnothing 1=s(0)=\varnothing \cup \{ \varnothing \} = \{ \varnothing \}=\{0\} 2=\{0,1\} ...

f:\varnothing \rightarrow A exists since for every x in the domain \varnothing there is a unique y in the codomain A such that (x,y) \in \varnothing . It is a vacuously truth since there are not any x in the domain. No f:A \rightarrow \varnothing exists by definition of a function since no...

I think a http://en.wikipedia.org/wiki/Construction_of_the_real_numbers" would do. "A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be...

For instance, a forgetful functor U:R Mod \rightarrow Set has a left adjoint F such that X \mapsto FX, which generates a free R-module using a basis set. So a free construction generates an algebraic structure using a set. http://en.wikipedia.org/wiki/Forgetful_functor" is the description...

(1) "Elements of Set Theory" by Herbert B. Enderton (2) "A Mathematical Introduction to Logic" by Herbert B. Enderton (1) includes a naive set theory and an introduction of an axiomatic set theory. Enderton's expository style of (1) is accessible to students without having much mathematical...

O(n) time algorithm seems possible. You have n integers stored in an array, each having an index. When going through each index, 1. if it is an even number, store it with an index. Otherwise go through next index. 2. Compare it with the stored value if it is an even integer, and update...

Z is the initial object of category of rings with morphism f:Z->S satisfying f(1z) = 1s (1z is the mulitplicative identity of Z and 1s is the multiplicative identity of a ring S. That means, a ring homomorphism f from Z to any ring is unique as long as f:Z->S satisfying f(1z) = 1s.

(f(x)) or <f(x)> (which one is standard?) denotes the ideal of Q[x] (polynomials with coefficients Q) generated by a subset of Q[x], which is f(x). It is the set of all elements whose form are r*f(x) where r belongs to Q[x].

The smallest ideal containing above f(x) is probably (f(x)). The maximal ideal containing f(x) is both (x^2+x-1) and (x-3). You can find the ideals in between.

\forall x\alpha\rightarrow\alpha^{x}_{t}, where t is substituable for x in \alpha.\alpha^{x}_{t} is the expression obtained from the fomula \alpha by replacing the variable x, whenever it occurs free in \alpha, by the term t (see "substitution section" of First-order logic (wiki). \forall...

http://en.wikipedia.org/wiki/Soundness" If Γ ⊢ P, then Γ ⊨ P. Soundness tells that deductions lead only to "correct" conclusions. If the deductive system is not sound, a proof might lead to a wrong conclusion.