I am trying to understand the set theoretic definition of functions along with its informal one, and would appreciate your comments.
Firstly, Given sets $A$ and $B:$
Relations: Any subset of $A \times B,$ that is $R \subseteq A \times B.$
When an ordered pair $(a,b)$ from $A \times B$ belongs to $R$, i.e. $(a,b) \in R$, then we write $a R b$ — pronounced as $a$ is related to $b$ under $R$
Functions: Relations with special properties as follows:
-
Each $a \in A$ is related to at least one $b \in B$ under $F.$ That is, for each $a \in A,$ there exists at least one $b \in B$ to which $a$ is related under $F$. In Quantifiers : For each $a \in A, \exists b \in B$ such that their ordered pair $(a,b) \in F.$
-
Each $a \in A$ is related to at most one $b \in B$ under $F.$ That is, for each $a \in A,$ there exists at most one $b \in B$ to which $a$ is related under $F.$
Together we have,
Each $a \in A$ is related to one and only one $b \in B$ under $F.$ That is, for each $a \in A,$ there exists one and only one $b \in B$ to which $a$ is related under $F.$ In Quantifiers : For each $a \in A, \exists! b \in B$ such that their ordered pair $(a,b) \in F.$
We can also simply say that $F \subseteq A \times B$ is a function if, and only if, $F$ contains exactly one ordered pair $(a,b)$ for each $a$.
Also,
Function (Informal) : A function from set $A$ to set $B,$ written as $f: A\to B,$ is an assignment which assigns to each element of $A$, exactly one element of $B.$ Meaning, for each $a \in A,$ there exists exactly one $b \in B$ which is assigned to $a$ by $f.$
$f(a)$ simply means the element of $B$ assigned to $a$ by $f.$
In Quantifiers: $\forall a \in A, \exists! b \in B,$ such that $b$ is assigned to $a$ by $f,$ that is $b=f(a)$
Would be great if you could assure me the veracity of my understanding.
I have another question: Why do books always define ideas a bit differently ? Is it really a matter of preference or some other advantage ? Like one book simply defined $F \subseteq A \times B$ a function iff $Dom(F)=A$ and $(a,b) \land (a,c) \in F \implies b=c.$ I know that these two conditions also establish the same idea, but I was just wondering, like when a person tries to understand a new concept for the first time, one may come across authors defining the supposed fundamental ideas slightly differently, but still equivalently.
Best Answer
Everything you wrote looks good to me.
Maybe not really an answer on "why" in general, but a note on this particular difference between these two definitions of function:
The symbol $\operatorname{Dom}$ is just a representation of the same idea that given an $a$ there exists a $b$ with $(a,b) \in F$, expressed as a set of $a$ values rather than a predicate on variable $a$. If the text has already introduced $\operatorname{Dom}$ (because it will be useful again), might as well use it in the definition of function.
Your first definition has "for each $a \in A$, there exists one and only one $b \in B$ to which $a$ is related under $F$", or using predicates, $\forall a \in A, \exists! b \in B, (a,b) \in F$. Your second definition has $(a,b) \in F \land (a,c) \in F \implies b = c$. This is implicitly universally quantified, actually saying: $\forall a \forall b \forall c, (a,b) \in F \land (a,c) \in F \implies b = c$.
But in formal predicate logic, the quantifier $\exists!$ is actually an abbreviation. For a predicate $P$ of one free variable, the notation $\exists! v, P(v)$ means exactly:
$$ (\exists x, P(x)) \land (\forall x \forall y, P(x) \land P(y) \implies x=y) $$
(This uses only fundamental predicate logic. Attempts to interpret "exactly one" or "only one" in the most literal way using a count of elements satisfying the predicate would be more complicated or impossible, presumes an understanding of at least natural numbers, etc.)
So the statement $(a,b) \in F \land (a,c) \in F \implies b=c$ is just writing out the "at most one" piece of the $\exists!$ idea in a standard way.
Then the first definition is one small abstraction away from the most fundamental predicate logic definition by use of the abbreviation $\operatorname{Dom}$; the second definition is one small abstraction away by use of the symbol $\exists!$. I'd call these differences just "rephrases" of essentially the same ideas. I suppose the sameness might not be obvious at first, and part of this is how familiar the reader is with the tools and conventions being used to state the ideas.
Of course, beyond this example, some differences between texts might be similar "rephrases". Some could involve a less trivial reason for equivalence of related statements. Some could introduce ideas in a different order, which requires different justifications or proofs entirely, even when the results are the same. There are different degrees of similarity in exactly what's being said and how it's being presented.