# implication vs equivalence when solving equations

I remember we were taught in high school (Eastern Europe) the difference between implication ($\Rightarrow$) and equivalence ($\Leftrightarrow$) and were instructed, when solving equations to be mindful to always proceed with equivalances, not implications so that we may trace our way backwards (since otherwise any solution which we arrived at might not satisfy the original equation we set out to solve).

So we had to write things like:

$x^2=y^2 \Leftrightarrow |x|=|y|$ (which makes perfect sense)

and (IIRC):

$x=y \Leftrightarrow x^2=y^2 \land x\cdot y\ge0$ (which also makes sense, but only if someone explains it to you)

Fast-forward twenty years I am revisiting old subjects to tutor my son. To my astonishment, I notice that people (e.g. Wikipedia or Wolfram MathWorld) don't bother using any kind of notation when they move from one equation to the next.

E.g. I see things like the following (while deriving the equation of an ellipsis):

$\sqrt{(x+c)^2+y^2} = 2a - \sqrt{(x-c)^2+y^2}$

immediately followed by:

$(x+c)^2+y^2 = 4a^2 - 4a\sqrt{(x-c)^2+y^2} + (x-c)^2+y^2$

which seems sloppy in my view as I would expect it to be followed by:

$(x+c)^2+y^2 = 4a^2 - 4a\sqrt{(x-c)^2+y^2} + (x-c)^2+y^2 \land 2a \ge \sqrt{(x-c)^2+y^2}$

Has there been a trend in recent years to de-emphasize the distinction between implication and equivalence when solving education or maybe this kind of notation was never really used in the West at the secondary education level? Or maybe it's not a good idea after all to insist on these nuances?

• In my experience in the U.S.: such logical connectors are not used in secondary school (although they should be). At the college level, it seems more common to use the implication symbol than the various conjunctions you show above. E.g., in the last example, it seems more natural to just use implication than to include the compound conjunction statement. I have never heard of a prohibition against correctly-used implication statements in this way. – Daniel R. Collins Mar 25 '18 at 2:35
• Not using logical connectors in some way is very wrong in my opinion; but I would not necessarily use implication. In some cases, it is easier to use implication to reduce the scope of possible solutions, you simply have to check back when the scope is sufficiently reduce to check the potential solutions to see if they are genuine solutions. – Benoît Kloeckner Mar 25 '18 at 20:06
• If you want to get all solutions to an equation, you should use equivalences. But if you simply want to prove the existence of a solution, implications would suffice. – Dan Christensen Mar 26 '18 at 14:01
• I agree with @DanChristensen, it's the fact that logical connectives arn't used in American schools and this becomes a bit a problem when I teach them in college. A lot of student really don't know the difference between 'if' and 'if and only if'. Since a lot of the teaching materials online are created by people from the American system, I think that's what you're seeing. – Nate Bade Mar 29 '18 at 6:09
• I can't help with a detailed response, but I can confirm that they have payed attention to this in Hungary recently (I finished school in 2012), but as a teacher in London now I can see that nobody cares about it. So I'd say the shift is more likely to be region-based as opposed to time-based. Again, I can't say this for certain, only my idea. – user16021 Mar 31 '18 at 8:51

## 2 Answers

Not that I mind the slight rant in the other answer, but

... maybe this kind of notation was never really used in the West at the secondary education level

seems closer to the truth. Those arrows are relatively modern inventions, from around 1920 onwards. I'm not aware of any widely used alternative notation in the mathematics and physics literature pre 1920. Even today I don't see people using $$\implies$$ or $$\iff$$ that often, neither in research literature nor undergraduate nor graduate books. One either uses words like "hence" "therefore" "equivalently" etc. or one doesn't write anything, assuming the reader will fill in the details. So I doubt this was ever standard in secondary education. (But I'd be interested in being proven wrong.)

Concerning your second question, I would personally rather emphasise the difference between "implies" and "is equivalent", but I'm not sure at what level and to which extent.

In the U.S. we forgot the spirit of mathematics as we embraced the race toward solution-finding in the mid-1900's (think NASA and military-supported efforts based on those branches' interests). In the business of finding a solution, you are allowed to be sloppy with these implications a bit more and then the idea is to sharpen their understanding of the logic later on at university.

For what it's worth I totally disagree with this way of approaching mathematics, but I can't help but think this is exactly what has happened. Once it becomes normal, it takes over, and becomes ubiquitous.

You're right, the lack of distinction is terrible. But it's fairly clear why there is no distinction if you think about the way a culture values mathematics.

• In the U.S. we forgot the spirit of mathematics as we embraced the race toward solution-finding in the mid-1900's (think NASA and military-supported efforts based on those branches' interests). Do you have evidence for this? This seems extremely implausible to me. – Ben Crowell May 4 '19 at 14:57
• @Ben Crowell: Indeed, as I suspect most anyone here from the U.S. is aware, the NASA era (late 1950s to mid 1970s) was when logic and rigor were at their highest point in U.S. school mathematics, at least in spirit (in practice, of course, the result was primarily an increase in formalism). For cheyne, see my answer to Where can I find primary sources from the New Math movement in the 60s? for some relevant literature. – Dave L Renfro May 4 '19 at 15:13
• @BenCrowell Here is my evidence: we went from 0-1 calculus courses before the early to mid 1900’s and now math majors in undergrad have 3-5 calculus courses in their major. Calculus is not the theory of formal mathematics, it’s the mathematics of “calculating stuff”. Going back into the 1500-1700’s in the history of calculus, when you introduce this topic as the bulk of your math major you are prioritizing calculations over logic and mathematical thought. I’m not really an expert on this, so I’m happy for someone to tell me this isn’t why that change occurred without suggesting a book. – cheyne May 5 '19 at 13:06