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What is the general definition and the exact sequence ?

Definition: A relation F from A to B is a function from A to B if and only if it meets both of the following conditions:

  1. Each element in the domain of F is paired with just one element in the range, i.e., from ∈ F and ∈ F follows that b = c.

  2. The domain of F is equal to A, domF = A.

(Source)(http://people.umass.edu/partee/NZ_2006/Set%20Theory%20Basics.pdf)

Definition :A function f whose domain is the set of a11 positive integers 1, 2, 3, . . . is called an infinite sequence. The function value f(n) is called the nth term of the sequence.in other words $f:\mathbb{N}\to A$

(Source)(http://www.matematica.net/portal/e-books/Apostol%20-%20CALCULUS%20-%20VOLUME%201%20-%20One-Variable%20Calculus,%20with%20an%20Introduction%20to%20Linear%20Algebra.pdf)

now :

1-) let :$a_n=\frac{1}{n-3}$

Is this($a_n=\frac{1}{n-3}$) a sequence?

2-) let A function $f$ whose $f:\{0\}∪\mathbb{N}\to A$

Is this($f$) a sequence?

3-) let A function $g$ whose $g: \{1,2,3,...,n\} \to A$

Is this($g$) a sequence?

and etc.

What is the general definition and the exact sequence ?????

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    There seem to be some (OCR?) errors in the quoted definitions, but I suppose we all can fill the gaps ... (and please don't expect us to flip through 600+ pages to check your quotes) – 2017-02-11

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1-) No, the domain of this function does not include $n=3$ so it does not satisfy your definition of sequence.

2-) No, the domain of this function is $\{0\}\cup\mathbb{N}$, whereas a sequence is required to have domain $\mathbb{N}$. The domain is not right, so it is not a sequence, but you could restrict it to nonnegative numbers to get a sequence.

3-) No, this is not a sequence. A sequence is required to have domain $\mathbb{N}$, not the smaller set $\{1,2,3,\dotsc,n\}$.

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    Actually, 3 is lso a "No". - $\{1,2,3,\ldots, n\}\ne\Bbb N$. – 2017-02-11
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    @HagenvonEitzen yep! Missed that, thanks. – 2017-02-11
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    but : in Page 98 (https://math.kennesaw.edu/~jlewin/CUP/marielle.pdf) It is said sequence :(( – 2017-02-11
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    @Almot1960 is there a problem with something on page 98? I didn't see anything, can you clarify what the issue is? – 2017-02-11
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    We could define f (n) = √n − 6 for every integer n ≥ 6.((Source)(https://math.kennesaw.edu/~jlewin/CUP/marielle.pdf)(Page 98) but set (n ≥ 6) $\neq \mathbb{N}$ – 2017-02-11
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    @Almot1960 On page 98 is given two definitions. The second one starts after the phrase "A slightly more general and more useful notion of a sequence". $\sqrt(n-6)$ is a sequence under the more general definition which doesn't require that the domain be all of $\mathbb{N}$, but not under the less general definition which you quoted in your question. – 2017-02-11
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    What is the general definition a sequence? – 2017-02-11
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    @Almot1960 definitions are a matter of convention and personal taste so different authors may have different definitions, but the one you gave is a good one: a sequence is a function with domain $\mathbb{N}$. It doesn't really entail a loss of generality vis-a-vis the more general definition, since you can just reindex. – 2017-02-11
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    sequence:A function f whose domain is Countable set . This definition is good? – 2017-02-11
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    @Almot1960 countable? Or countably infinite? I'd prefer the latter, "countable" also includes finite sets. Sets from countable finite sets could perhaps be called "finite sequences", but I wouldn't use the word "sequence$ without qualification. Note that countably infinite sets have bijections with $\mathbb{N}$, so sequences indexed by countably sequences can be put in correspondence with sequences indexed by $\mathbb{N}$. They are equivalent definitions. – 2017-02-11
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    Please edit:))Thank you. – 2017-02-11