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mpitluk
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For any set S, the natural numbers N and function f, if f : S → N is injective but not surjective, is S finite?
mpitluk said:For any set S, the natural numbers N and function f, if f : S → N is injective but not surjective, is S finite?
mpitluk said:Sorry, I'm not sure what that tells me. I have VERY little mathematics training, but ended up taking a math-logic course heavy on notation and dependent on higher-math knowledge.
It seems to me what you are saying, though I am probably dead wrong, is that the set of odd naturals is in a bijection with the naturals. And thus, they have the same cardinality. But, I'm asking about a case in which S is not surjective.
mpitluk said:Wow. I see where I went wrong. What I meant to ask, while trying to get the notation down, was: if you have a set A that doesn't have a bijection with a set S such that |S| = |N|, then is A finite? It seems to me it would be (by definition, really).
I was referring to the following definition: for a set S and the set of naturals N, if |S| < |N|, then is S finite. I see where I went wrong. I am just trying to define a finite set using the terms "bijection," "surjection," and "injection."DonAntonio said:And "by definition" of what?
DonAntonio
DonAntonio said:No. S could be, say the set of all real numbers, which cannot mapped bijectively with the naturals...
DonAntonio
mpitluk said:I was referring to the following definition: for a set S and the set of naturals N, if |S| < |N|, then is S finite. I see where I went wrong. I am just trying to define a finite set using the terms "bijection," "surjection," and "injection."
Might this be right: if for every mapping f between S and N, f : S → N is not surjective, then S is finite.
DonAntonio said:I guess that could work, but why do you seem to enjoy making things messy? Go to the following definition:
"A set S is finite iff EVERY proper subset of S has a cardinality strictly smaller than that of S".
Voila
DonAntonio
A "Yes or No? Injection into the Naturals finite" refers to a mathematical concept where a set of natural numbers is either injected (mapped) into a set of yes or no values or vice versa.
This concept can be represented using a function, where the input is a natural number and the output is a yes or no value. Alternatively, it can also be represented using a truth table, where the natural numbers are listed as inputs and the corresponding yes or no values are listed as outputs.
Studying this concept can help in understanding the relationship between two sets and how they can be mapped to one another. It also has applications in computer science, logic, and decision-making.
Yes, there are limitations to this concept. It can only be applied when the set of natural numbers is finite. Additionally, the mapping must be one-to-one, meaning each natural number must correspond to a unique yes or no value and vice versa.
An example of this concept is mapping the set of even numbers to the set of yes and odd numbers to the set of no. Another example is mapping the set of prime numbers to yes and non-prime numbers to no.