Another maths/logic thread! help please!

megakid

megakid

megakid

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Hey

I've come unstuck with a question I've got for one of my module exercises.
I'm more looking for advice rather than a straight out answer, just to find out where I'm going wrong.

I've been asked to define, for the set of all such functions, an injective partial function.
I've defined the complete injective function as:

X >--> Y == { f: X<->Y | Ax:X; y1, y2:Y . x|-> y1 elementof f ^ x|-> y2 elementof f => y1 = y2 }

I'm stuck on the partial injective function - How do I define the above but with extra elements in X that arent mapped. I dont know how to show that.

Any help would be fantastic. And to those who dont understand what the hell I'm talking about, I'm sorry for dampening your Monday night ;)
 
After a bit more thought I *think* i've solved it -

X >-|-> Y == { f: X<->Y | Ax:dom(f); y1, y2:range(f) . x|-> y1 elementof f ^ x|-> y2 elementof f => y1 = y2 }
 
Yeh sorry - just realised that makes no sense what-so-ever ... I'm just trying to define a partial injective function.

I've simplified it slightly:

X >--> Y == { f: X <--> Y | A[all] r,s:X . (f(r) = f(s)) => (r=s) }

Which means my version of a partial function would be:

X >-|-> Y == { f:X <--> Y | A[all] r,s:dom(f) . (f(r) = f(s)) => (r=s) }
 
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The difference between a total and partial function is that:

EDITED OUT TO AVOID CONFUSION :P

This has nothing to do with injection which is where each mapped element in X is mapped to a unique Y element. eg. If 2 is mapped to 4 then nothing else can be mapped to 4.

I have no issues with the concepts (I dont think?) - it's just the logic predicates that I'm writing to describe them.

EDIT wait - i think i've just completely confused myself...you might be right...I'll get back to you lol

EDIT2 - edited for correctness - completely wrong the first time.
 
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Lagz said:
An injective function is simply a function where f(a) = f(b) => a=b.
It is indeed nothing to do with partial and total functions.

A partial function is a function which MAY have some a such that f(a) is not defined. Total functions are actually a subset of partial functions. It is not true that a partial function MUST have some a such that f(a) is undefined.

Since an injective function doesn't say anything about the type of function and since a partial function is essentially any function, saying the function is partial adds nothing to the definition.
Click to expand...

I see - I thought partial functions were a strict subset of total functions.
 
Yes sorry, my mind is in overdrive. Total functions are a strict subset of partial functions.

Actually have I got something completely wrong here -

Your definition of a partial function, which MAY have some a such that f(a) is not defined, bugs me.

X <--> Y

With a partial function, from your definition, this means that an element in X is unmapped - It says nothing about Y.

We've been taught differently here :S Says here straight off my Z/schema notes that a partial function (X -|-> Y) maps everything from the first set X, to just SOME things in Y.
 
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Lagz said:
It doesn't mean an element in X is unmapped. It just means an element in X MAY be unmapped.
Click to expand...
Yep - I understand that.

Lagz said:
And no it doesn't really say anything about Y (apart from that any X which is mapped must point to a single value in Y - I didn't say this explicitly. . I assumed it was obvious :-)).
Click to expand...

Hmmm doesnt that mean it's injective then? :)
 
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