[formal-methods] Injective, Surjective, and Bijective functions

[d p chang]

Mikael Vejdemo-Johansson <mik at stanford.edu> writes:

> On Jun 28, 2009, at 4:58 AM, d p chang wrote:
>> Mikael Vejdemo-Johansson <mik at stanford.edu> writes:
>>>
>>> f is injective iff f(x) = f(y) implies x = y.
>>
>> hmmm... i was thinking that one would have to prove that f(x) was
>> defined as well (ie, f was 'closed' over Z). 
>
> If f is a function from A to B, we assume by definition that f(a) is  
> defined for every a in A, and takes on a unique value.

ok. i guess i was over thinking something here.

> There is the concept of partial function, but that's a completely
> different mess to get involved in.

i wasn't thinking partial, but more 'is it really defined at all'. now
i'll have to go back and check my understanding of partial vs total, but
i guess anything non-total is partial.

anyway, i was blocking on the fact that i didnt think i could 'prove' if
(*2) was defined over Z (regardless of intuition). i had some handwavy
thoughts about contradiction (ie, suppose that it wasn't for some x)
which is how i got to *2 == x + x. at this point, i should have realized
your first point above about 'by definition' but spun my wheels thinking
about proving Z was a ring (i think the right term) rather than
remembering it was 'defined' to be one.

clearly, my math/vocabulary/thinking is definitely 'applied' rather than
'theory' :-)

\p
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