Negating Definition of Function: A=>B

[solakis1]

We know that we define a function from the set A to the set B ,denoted by f: A=>B
iff:

1) f is a subset of AxB

2) For every a belonging to A ,there exists a unique b belonging to B,such that (a,b) belongs to f

In trying now to negate the above definition i got stuck ,particularly in negating statement (2).

Because i had to first correctly formalize the statement and then negate it by appling the appropriate laws of logic.

 

[Ackbach]

I would work from the outside in. To go to symbols here,
$$2) \; (\forall a \in A)(\exists \text{ unique }b \in B)|(a,b) \in f.$$
The tricky part, it seems to me, is the uniqueness. How can you get that into symbols?

Take a simpler case: try translating the statement "There exists a unique $x$ satisfying property $P$" into symbols. How can you get at the uniqueness?

 

[solakis1]

I would work from the outside in. To go to symbols here,
$$2) \; (\forall a \in A)(\exists \text{ unique }b \in B)|(a,b) \in f.$$
The tricky part, it seems to me, is the uniqueness. How can you get that into symbols?

Take a simpler case: try translating the statement "There exists a unique $x$ satisfying property $P$" into symbols. How can you get at the uniqueness?

Thanks for your help,should not the formula that you suggest be first of all a well formed formula?

 

[Ackbach]

Thanks for your help,should not the formula that you suggest be first of all a well formed formula?

Absolutely, it should. I was not worrying about syntax overmuch. How would you get it to be a wff?

 

[solakis1]

Absolutely, it should. I was not worrying about syntax overmuch. How would you get it to be a wff?

We have to follow the rules

 

[Ackbach]

We have to follow the rules

Well, yes. That's not exactly what I was driving at. Why don't you take the formula I gave above, and turn it into a wff? What would be that specific result?

 

[MarkFL]

Adrian, I caution you with a tale of "Brer Rabbit:"

 

[solakis1]

Well, yes. That's not exactly what I was driving at. Why don't you take the formula I gave above, and turn it into a wff? What would be that specific result?

And the rule is:

1)If G is an n place predicate symbol and $$t_{1},t_{2}...t_{n}$$ are terms (not necessarily distinct),then $$G(t_{1},t_{2}...t_{n})$$ is a formula (an atomicformula)
2) If P and Q are formulas,then (P=>Q),(PvQ),(P^Q) ARE formulas.
3)if P is a formula then ~P is a formula.
4)If P is a formula and u is a variable ,then $$\forall u P$$ is a formula
5) If P is a formula and u is a variable ,then $$\exists u P$$ is a formula
6) Only strings (= a finite sequence of symbols) are formulas,and a string is a formula only if its being so follows from (1).(2),(3),(4).or (5)

According to (4) and (5) in the above definition strings of the form $$\forall a\in A$$, $$\exists b\in B$$ are not correct,but strings of the form $$\forall a(a\in A)$$ ,$$\exists b(b\in B)$$ are

Do you agree?

 

[I like Serena]

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