I was reading through Mathematical Foundations of computing (preliminary course note by keith Schwarz pg 15 ) and noticed there was a definition for sets of positive natural numbers $\mathbb{N}^+ = \{ 1,2,3...\}$, noting that $0 \notin \mathbb{N} ^+ $ and I also read in Mathematics for computer science ( by Eric Lehma pg 14) that $\mathbb{Z}^+$ represents positive integers having the same set $ \{1,2,3,\dotsc\}$. So my question is, are they the same set? Is there any difference between the set of positive natural numbers and positive integers?
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1$\begingroup$ It's just a matter of notation. The author should make clear what notation is being used at the beginning of the book. I personally prefer to write e.g. $\{0,1,2,\ldots\}$ or $\{1,2,\ldots\}$ to avoid ambiguity or say "nonnegative integer" or "positive integer". $\endgroup$– Math1000Jan 8, 2020 at 13:24
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1$\begingroup$ If you want to be horribly pedantic... you could argue that the $1$ that exists in $\Bbb N$ is officially equal to the set $\{\{\}\}$ as it is defined in the Von Neumann construction of the natural numbers while the $1$ in $\Bbb Z^+$ is the equivalence class $\{(1,0),(2,1),(3,2),\dots\}$... much like how the element $1$ from the real numbers is technically a dedekind cut and the element $1$ from the complex numbers is $1+0i$ and so on... Now... despite each of these $1$'s being defined in different ways, they will all behave similarly and we change which we are referring to at whim $\endgroup$– JMoravitzJan 8, 2020 at 13:30
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2$\begingroup$ Does this answer your question? Is "$a + 0i$" in every way equal to just "$a$"? $\endgroup$– JMoravitzJan 8, 2020 at 13:35
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1$\begingroup$ For all intents and purposes, the sets are the same. Practically, they are the same. There exist canonical embeddings from one to the other. Pedantically, one might argue that they are defined in different ways and thus are not equal. I highly recommend reading the linked paper in the linked answer "what does it mean for two things to be 'equal'" $\endgroup$– JMoravitzJan 8, 2020 at 13:49
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1$\begingroup$ I find that response, to express it in your choice of words, much more sententious and turgid than JMoravitz' comment, which is substantially correct and appropriate given the tags. $\endgroup$– ThorgottJan 8, 2020 at 13:50
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Yes, the two sets are equivalent.
The natural numbers and the integers are constructed by very different methods, so by some arguments they are not even the same mathematical object. (For instance, one might argue that integers are the quotient set of pairs of natural numbers under an equivalence relation!)
However, at the end of the section of those constructions, authors traditionally note (or prove) that all such constructions of number systems are isomorphic to each other, and so they define completely abstract sets $\mathbb N$, $\mathbb Z$, $\mathbb Q$,and $\mathbb R$ such that we can legitimately say that $\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R$. In that sense (which is the typical sense), it would be completely legitimate to say that $\mathbb N^+=\mathbb Z^+$.
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$\begingroup$ Thank you very much Matthew, I think i can complete my assignment now. $\endgroup$ Jan 8, 2020 at 13:49