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An amateur mathematician, I have developed DC Proof, an educational software program to introduce students to the basic methods of proof. In the style of most mathematics textbooks at the undergraduate level, its simplified rules of logic and set theory are loosely based on standard FOL and ZFC. For more information, a video demo and free download, visit my website at http://www.dcproof.com.

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16h
revised Need help with a fundamental theorem of finite arithmetic
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2d
comment Notation for Partial Functions
It doesn't seem that you can formally define $f(x)$ in isolation, even for total functions.
2d
comment Notation for Partial Functions
In a formal proof, you won't see a definition anything like $\forall x\in A: [f(x) \iff ... ]$. The $f(x)$ would have to be embedded in logical expression, e.g. $f(x)=y.$
2d
comment Notation for Partial Functions
Even for total functions of 1 variable, the standard approach for a long time (centuries?) has been to define $f(x)=y \iff (x,y) \in f$. Do you have a problem with that? I am proposing to do the same thing with partial functions. It seems to work.
Apr
21
comment Notation for Partial Functions
I don't think we formally define $f(x)$ all on its own even for total functions. Rather, we define either $f(x)=y$ or $f(x)\in$ some set.
Apr
21
comment Notation for Partial Functions
As I have pointed out elsewhere, even for a total function $f:A\to B$, we define $f(x)=y \iff (x,y)\in f$. Quite a mainstream notion as it turns out. See en.wikipedia.org/wiki/Function_(mathematics)#Notation
Apr
21
comment Notation for Partial Functions
Admittedly, it would be difficult to attach a precise meaning in everyday language to $f(x)=y$ in that case, but no more so than attaching a meaning to your $f^{\bot}$ construction. If $f$ isn't defined at $x$, then $(x,y)\in f$ is false, and, by definition, so is $f(x)=y$. $f(x)\in B$ would also be false.
Apr
21
comment Notation for Partial Functions
Sounds like a lot of bother for such a simple idea.
Apr
21
comment Notation for Partial Functions
Interesting to know that there may be no widely accepted formalism for partial functions.
Apr
20
comment Notation for Partial Functions
@dtldarek Since the latter "definition" can be derived from the former (not sure the other way), we should probably concentrate on the equality, $f(x)=y \iff (x,y)\in f$. Given that images under $f$ are unique, this shouldn't cause a problem.
Apr
20
revised Notation for Partial Functions
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20
revised Notation for Partial Functions
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20
revised Notation for Partial Functions
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Apr
20
revised Notation for Partial Functions
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Apr
20
comment Notation for Partial Functions
Thanks, but how commonly used is this convention? Can it truly be something novel as suggested by dtldarek? The few definitions of partial functions that I have seen don't explicitly define the very useful $f(x)$ notation .
Apr
20
comment Notation for Partial Functions
$f(x)$ in isolation could be seen as a sentence fragment. It only makes sense embedded in an syntactically corrected sentence, e.g. $f(x)\in B$. And yes, this is true if and only if $\exists y:[y\in B \land (x,y)\in f].$ I know that others have trouble with this. That is why I posted this question. My definition just seemed so reasonable to me. And it doesn't seem to lead to any contradictory or unexpected results.
Apr
20
comment Notation for Partial Functions
@dtldarek Judging by Berci's response, it isn't all that novel. It makes doing formal proofs about partial functions much easier. Note that this definition also works for total functions.
Apr
20
comment Notation for Partial Functions
@dtldarek If $D$ is the subset of $A$ such that $\forall x:[x\in D \iff x\in A\land \exists y:[y\in B \land (x,y)\in f]]$, i.e. $D$ is the domain on which $f$ is a total function, then it can be shown using my latter definition that $\forall x\in A :[f(x)\in B \iff x\in D].$
Apr
20
comment Notation for Partial Functions
@dtldarek A stronger argument than any supposed meaninglessness would be a demonstration that my definition would lead to some contradiction, e.g. a counterexample leading to $f(x)\in B \land f(x)\notin B$.
Apr
19
comment Notation for Partial Functions
The latter is what I have usually seen. How common a convention is the former?