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 \(\displaystyle t_{1},t_{2}...t_{n}\) are terms (not necessarily distinct),then \(\displaystyle G(t_{1},t_{2}...t_{n})\) 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 \(\displaystyle \forall u P\) is a formula
5) If P is a formula and u is a variable ,then \(\displaystyle \exists u P\) 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 \(\displaystyle \forall a\in A\), \(\displaystyle \exists b\in B\) are not correct,but strings of the form \(\displaystyle \forall a(a\in A)\) ,\(\displaystyle \exists b(b\in B)\) are

Do you agree?

 

[I like Serena]

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