What is the meaning of the weird symbol in 'Mathematical Logic'?

[omoplata]

What's the symbol in the attached image, that looks like a right pointing arrow, but with a short perpendicular arrow at the base?

The book is 'Mathematical Logic' by Cori and Lascar.

They don't explain what it is.

Maybe it's the symbol for function?

 

[sfs01]

Assuming you mean this one: \neg That's logical not/negation.

 

[omoplata]

No, I mean this one
\mapsto
Well, I guess it's "maps to" since the latex code is "\mapsto".
What does "closed under the operation" mean? They haven't defined it.

 

[Fredrik]

Yes, "maps to". For example, the function f defined by f(x)=x² for all x can also be written as x\mapsto x^2. (Most people just write x², which is strictly speaking incorrect. That expression represents a member of the range).

When they say "closed under the operations...", they just mean that the things on the right are members of the set \mathcal F. (That's a strange looking "F". It looks more like a "P"). Examples of how to use the word "closed": The set of integers is closed under addition. The set of positive real numbers is closed under multiplication.

 

[micromass]

Hi omoplata!

I'm not sure how much mathematics you know, so ask if something is not clear...

Every operation, like \wedge can be seen as a function:

\mathcal{W}(A)\times \mathcal{W}(A)\rightarrow \mathcal{W}(A):(F,G)\rightarrow F\wedge G

So an operation is actually a function that takes two strings of symbols to a string with \wedge between it. Now, usually a function is written as

\mathcal{W}(A)\times \mathcal{W}(A)\rightarrow \mathcal{W}(A):(F,G)\mapsto F\wedge G

where the \mapsto is just a notation to denote that (F,G) is being sent to F\wedge G.

(I actually find the mapsto symbol to be incredibly ugly so I never use it, even if it is standard and advisable to do so)

 

[omoplata]

Thanks for the explanations. That really helped.

 

[disregardthat]

It doesn't make sense to say that a set is closed under a operation if the operation is well-defined (which is a prerequisite for being an operation in the first place) on the set. It can make sense if an operation is defined on a set, and then saying that some subset is closed under the induced operation.

I have found that the maps-to notation is an effective method to define many functions when making large diagrams with arrows. It makes a clear representation as well.

 

[omoplata]

So, the operations in this case are \neg, \wedge, \vee, \Rightarrow, \Leftrightarrow, right? The only way they can be defined is through truth tables, right. They haven't defined them yet. They've just stated that there are five operations, and that the set \mathcal{F} is the smallest subset of \mathcal{W(A)} those operations are closed under. So is it 'legal'?

 

[micromass]

So, the operations in this case are \neg, \wedge, \vee, \Rightarrow, \Leftrightarrow, right? The only way they can be defined is through truth tables, right. They haven't defined them yet. They've just stated that there are five operations, and that the set \mathcal{F} is the smallest subset of \mathcal{W(A)} those operations are closed under. So is it 'legal'?

It certainly is legal. We haven't given any meaning to formula's, so far we have just made a set \mathcal{W}(A) which contains certain strings of symbols.
I could as well make a set G that contains all words with letters a and b. The set would consist of

\{a,b,ab,ba,aba,baaab,bbaabbabbbabbabaaabaabaabba,...\}

We didn't give any meaning to the words yet, we just selected a set which contains certain symbols.