Simple question, set theory

I’m having quite a bit if trouble finding a straight answer on how one should properly denote the set of all even and all naturals respectively. What is the standard, universally understood way to signify these?

=================

2

 

The term natural number is ambiguous, unfortunately. For many of us, including me, it means non-negative integer, and the set of natural numbers is denoted by N\Bbb N or by ω\omega. For some it means positive integer, and some of those people use N\Bbb N to denote their notion of natural number, thereby making the notation ambiguous.
– Brian M. Scott
2 days ago

=================

2 Answers
2

=================

Natural Numbers are denoted N\mathbb{N}.

Even Numbers are denoted 2N2\mathbb{N}.

  

 

Okay, so natural numbers are represented as N\mathbb{N} and even numbers are represented as 2N2\mathbb{N}. But how can we prove the denotation of the even numbers set?
– Obinna Nwakwue
2 days ago

  

 

@ObinnaNwakwue What do you mean prove? Even numbers are of the form 2n2n, so they are 2N2\mathbb{N} by definition.
– Pax Kivimae
2 days ago

  

 

“But how can we prove the denotation of the even numbers set?” I do not know what this question means. Notation is convention. I can denote the set of even numbers as PinKhP34doω\omegafoo8P but nobody would understand what I meant it I did. In context most people would assume A+BA + B to mean {a+b|a∈A;b∈B}\{a+b|a \in A; b \in B\} and with almost no coaxing assume 2A2A to mean {2b|b∈A}\{2b|b \in A\}. It’s not a matter of proving anything. It’s a matter of saying “Let a∈PinKhP34doωfoo8Pa\in PinKhP34do\omega foo8P. Do you know what I mean by that”. “Um… no.” “Okay, Let a∈2Na \in 2\mathbb N. Better?”. “a is even?” “exactly”
– fleablood
2 days ago

  

 

Although, I’m not one hundred percent sure how universally 2N2\mathbb N is accepted. There’s nothing wrong with simply stating: “For this paper I am going to denote the set of even numbers as 2N2\mathbb N.” or “For this paper I am going to denote the set of even numbers as E\mathbb E.” Or ” “For this paper I am going to denote the set of even numbers as PinKhP34doωfoo8P\mathbb {PinKhP34do\omega foo8P}.”
– fleablood
2 days ago

  

 

I meant…wait, never mind, I might have a generalization of this.
– Obinna Nwakwue
2 days ago

Usually if n∈Nn\in \mathbb N ,  nN={α≡0 mod(n)|α∈N}\space n\mathbb N=\{\alpha\equiv 0\space mod (n) |\alpha \in \mathbb N\}. So 2N2\mathbb N is the set of all even numbers.(Positive obviously).

  

 

Che serate impegnative eh? (entrambi, anzi, tutti e tre)
– Giuseppe Bargagnati
2 days ago

  

 

@GiuseppeBargagnati Pappalettera
– Marco Lecci
2 days ago

  

 

I made a mistake in asking my question. It was supposed to ask the notation regarding even and odd naturals. 2N seems fair for evens…I was using N sub o and N sub e…is that workable?
– Chris Hohl
yesterday