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Good evening, I'm currently reading up on mathematical logic, and have came across quantifiers, specifically, the universal quantifier ($\forall$) and the existential quantifier ($\exists$), both of which I hope to incorporate into my high school differential equations homework.

A simple question is this : Find the general solution of the differential equation, $\frac{dy}{dx} = 1.$

We start with integrating both sides: $$\begin{align}\int{dy} &= \int{dx}\\ y &= x + C\end{align}$$ My first confusion (more of on differential equations) comes here : is the above true "for all" $C \in \mathbb{R}$, or "for some" $C \in \mathbb{R}$?

Then, my second confusion : must we write $\forall{C\in\mathbb{R}} : y = x + C$, or is it also valid to write $y = x + C~ \forall{C\in\mathbb{R}}$ (I.E. prefix vs suffix)?

Also, my third confusion : how would I express "Given $\frac{dy}{dx} = 1$" at the very start of my workings? Is it alright to say $\frac{dy}{dx} = 1 \implies \int{dy}=\int{dx}$? Or would it be better to use iff ($\iff$) instead?

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  • $\begingroup$ @AndréNicolas Edited, thanks! $\endgroup$
    – Yiyuan Lee
    Feb 8 '14 at 8:08
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    $\begingroup$ I suggest you use words. $\endgroup$ Feb 8 '14 at 8:10
  • $\begingroup$ For your second question: suffix is fine. After all, what right does anyone have to tell you what is valid or invalid notation? As long as its sufficiently unambiguous, its valid. $\endgroup$ Feb 8 '14 at 8:13
  • $\begingroup$ @user18921 I just wish that those who are marking my papers (most probably with mathematics as their profession) would accept my "syntax". Thanks! $\endgroup$
    – Yiyuan Lee
    Feb 8 '14 at 8:14
  • $\begingroup$ @LeeYiyuan, if its unambiguous syntax, you have a right to appeal their decision by going to a higher authority (in my opinion). However, it might be more effort than its worth (up to you). $\endgroup$ Feb 8 '14 at 8:17
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First off, I must say that the use of the quantifiers in this context is a bit excessive. It's not needed and perhaps even confusing to the reader.

Let me note that the quantifiers are used in two very distinct manners: formally and informally.

Typically, quantifiers are placed before your proposition, i.e. you have $$\forall x\ P(x),\ \ \ \ \text{or}\ \ \ \ \exists x\ P(x)$$ where $P$ is some logical proposition you are quantifying over. Informally though, it doesn't really matter and the symbols are just a shorthand for the words, i.e. it's quite common to see a phrase along the lines of $$P(x),\ \ \forall x$$ You are really just writing shorthand for the sentence "P(x), which holds for all $x$". As long as your intentions are clear, feel free to use them as you like.

Formally though, quantifiers follow very strict rules in their usage and placement. Keep in mind that logical sentences are constructed to be very rigid. So rigid in fact, that they can serve as inputs to programs which perform logical operations on them.

It is way beyond the scope of this answer to talk about the formal use of quantifiers, although there are plenty of resources available if you are interested. Almost all uses of the quantifiers in math (outside of a few select branches such as set theory and mathematical logic) are informal or semi-formal. Your use of them at this point will certainly be informal.

To answer your question directly, you should probably write something along the lines of: $$\exists C\ \forall x\ y(x) = \frac{x^2}{2} + C$$ Although to be perfectly honest, I would be rather surprised if I saw someone actually write that.

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  • $\begingroup$ I think it's more appropriate to abandon $\forall x$. Since, for example, $f(x)=x^2/2$, say, is a definition of a function. $\endgroup$
    – Yai0Phah
    Feb 8 '14 at 8:52
  • $\begingroup$ @Frank Perhaps you're right. But then again, I find it more appropriate to abandon both $\exists C$ and $\forall x$. $\endgroup$
    – EuYu
    Feb 8 '14 at 21:35
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Good question.

Firstly, here's how I would write the solution if I wanted to be "rigorous." Begin with a convention. Whenever $x$ is a variable, let us write $\mathbb{R}x$ for the set of all real numbers dependent on $x$. Now for the solution.

Let $I$ denote an open real interval, declare $x$ to be a variable of $I$, and suppose $y \in \mathbb{R}x$ is differentiable with respect to $x$. Then TFAE.

  1. $\dfrac{dy}{dx} = 1$
  2. $\exists C \in \mathbb{R} : y = x+C$

Now secondly, let me just say that my personal attempts at solving more advanced differential equations in a "logically explicit" way have been fraught with difficulty. So if you want to get good at solving DE's, I recommend not to get caught up in rigor and/or being logically explicit. There's simply no book or resource available to teach you how to solve DE's rigorously, so trying to be logically explicit in this field is (currently) a major handicap to learning the material.

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  • $\begingroup$ Usually it's unnecessary to solve DE's rigorously in some sense. It's tedious and sometimes impossible. Instead, we solve to some degree, then appeal to the existence and uniqueness theorem of Picard. $\endgroup$
    – Yai0Phah
    Feb 8 '14 at 8:39
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    $\begingroup$ What does it mean for "a real number to be dependent on $x$"? And if $y$ is just some real number, why is $\frac{dy}{dx}$ not just $0$? It seems to me like you are making a new notation (i.e. $\mathbb{R}x$) for something that already has a perfectly good notation: i.e. a function $f: \mathbb{R} \to \mathbb{R}$. $\endgroup$ Feb 8 '14 at 8:40
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    $\begingroup$ @RyanSullivant Ah, yeah, but we still consider the variables with constraints in our mind, I think. For example, in complex analysis, we call $w$-plane and $z$-plane and view $w,z$ are related by some $w=f(z)$. In some formal sense, modern mathematics destroys the concept of such variables and bases the function on set theory, but it's usually convenient for us to think them in the old sense. Another example is on the concept of total derivative in PDEs. $\endgroup$
    – Yai0Phah
    Feb 8 '14 at 8:48
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    $\begingroup$ I was not saying claiming that the old way can't be formalized. The modern way is preferred by, say, algebraists, in my opinion. When arguing with commutative diagrams, diagram-chasing, abstract nonsense, etc, it's useful to think in that way. $\endgroup$
    – Yai0Phah
    Feb 8 '14 at 9:00
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    $\begingroup$ @user18921 I have no problems with variables, but I think the $\mathbb{R}x$ is not a good notation (personally). It is confusing, because you are letting $x$ be a variable (often when people write $x \Box A$ or $A \Box x$ for some set $A$, element $x$ and operation $\Box$ they fix the element $x$ so that is clear what the sets $x \Box A$ or $A \Box x$ are). Also, as far as I can tell it is non-standard, and so limits the ability to communicate ideas to the mathematical community. Eq 2 is perfectly fine with me, and that is exactly what we write when we are saying $y$ is a function of $x$. $\endgroup$ Feb 9 '14 at 20:59

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