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.