B Set Theory Question -- Which one is correct?

  • B
  • Thread starter Thread starter Heisenberg7
  • Start date Start date
  • Tags Tags
    Logic
Heisenberg7
Messages
101
Reaction score
18
I can't decide which one is better to use. I know for a fact that the second one is correct, but I would like to know if I can use the first one too. Which one would you use?
$$\forall x \in \mathbb{Z} (\exists y \in \mathbb{Z} : x > y)$$
Or
$$\forall x \in \mathbb{Z} (\exists y \in \mathbb{Z} , x > y)$$
Is there a different way to group these expressions?
 
Mathematics news on Phys.org
I prefer the first version. The ':' means only 'such that' to me and is less ambiguous than ','. But I think this is just a matter of taste.
 
  • Like
Likes Heisenberg7 and SammyS
I sometimes use
$$
\substack{\forall\\x\in \mathbb{Z}}\quad\substack{\exists\\y\in \mathbb{Z}}\quad x>y
$$
to avoid exactly this question. The comma notation is quite unusual. Another parenthesis would be better
$$
\left(\forall\;x\in \mathbb{Z}\right)\;\left(\exists\;y\in \mathbb{Z}\right)\;x>y
$$
A textbook on logic normally doesn't use any of them. Logic has its own notations like ##\dashv.## I once saw how Russell dealt with set theory. It was barely readable.
 
  • Informative
  • Like
Likes mcastillo356 and Heisenberg7
I would write:
##\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\, [ x>y ] ##
##\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\, ( x>y ) ##

It is also OK to write something like:
##\forall x \in \mathbb{Z} \, [\, \exists y \in \mathbb{Z} \,\, ( x>y ) ] ##
##\forall x \in \mathbb{Z} \, (\, \exists y \in \mathbb{Z} \,\, ( x>y ) \,) ##

Since the expression ##x>y## is really short, it seems to me that it should also be fine to write:
##\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\,\,\,\, x>y ##


I haven't seen a symbol like ##:## used in logical statements (but maybe it is used commonly and I don't know it). Normally I think the symbol ##:## is widely used [in place of ##|## ] in defining specific sets. For example, the set of even integers ##E \subseteq \mathbb{Z}##:
##E=\{x \in \mathbb{Z}: \exists k \in \mathbb{Z} (x=2k) \}##
 
Last edited:
Not sure I should bump the thread for a small point. But I think I kind of get how the symbol ##:## seems pretty reasonable for use in logical statements (or representing predicates etc.). Though for longer expressions, personally I think it might be easier to use brackets (at least for me).

Regarding the quesion in OP, I think the original expression (the first one) as written is fine [though this is also mentioned in the very first reply]. I would say it would also be OK to write:
##\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\,:\,\, x>y ##
 
  • Like
Likes Heisenberg7
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top