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General Discussion & Feedback => Just Chat! => Topic started by: alancalverd on 21/10/2021 00:31:59

Title: Equal or equivalent?
Post by: alancalverd on 21/10/2021 00:31:59
Mathematically, and indeed in all scientific phraseology, equivalent (≡) is "stronger" than equal (=). If A ≡ B then all their properties match for all tests we can apply. The same terminology is used in engineering: an equivalent component will have exactly the same specification as the original but a different part number because, say, it comes from a different manufacturer or was made for a different customer. X = Y, on the other hand, may only be true for a limited range of variables, maybe only a single point where two curves intersect.

In sociology, the terms are used in precisely the opposite sense. "Equivalent but separate is not equal" is the  essence of all desegregation slogans, and we have an intuitive grasp of its meaning: equivalent implies difference, not similarity.

How come?
Title: Re: Equal or equivalent?
Post by: Eternal Student on 21/10/2021 01:25:33
Hi.

This is in the just chat section, so I don't have to be too serious.

In sociology, the terms are used in precisely the opposite sense.
    That's interesting.

Mathematically, and indeed in all scientific phraseology, equivalent (≡) is "stronger" than equal (=)
    Yes.   I'd rather say something about this.....
I think the word  "equal"  and the symbol   =      was used first.   There was a need to have a new symbol and name for a stronger condition where two things are identical under all values of the variables.  I never thought it was any more complicated than this.
    A brief glance through some articles about the history of mathematical symbols seem to suggest that ≡ was first used  to mean  "equal by definition",   something for which the symbol   :=  is now often used, although you can still see that  Ξ  and "mathematical equivalence" could be used to convey the same information.  It seems quite common that various people introduced some notation and over enough time some common sense and consistency prevailed.   Others continued to use the symbols that seemed most sensible or consistent with what we already had and the modern set of mathematical symbols emerged similar to way the English language evolved through common usage.

So, looking purely for the good reasons or logic that would make the symbol  Ξ  endure we have the following ideas (among others, I'm sure):
  1.  Three lines is more than just two lines.   So  ≡  is intuitively a stronger symbol than =.   
  2.  Just by looking at a small part of it you know it means the same as  =   but with more.       
  3.  It follows similar conventions to other symbols in common use  like    ≥  used on numbers  which implies more than just  >  and also  set theoretic notation   b2412c37ae749271a3e0a1cd7a4c80d3.gif   which means more than just   881a4938d25fa6cf3f70c825c33bafde.gif

Best Wishes.
Title: Re: Equal or equivalent?
Post by: alancalverd on 21/10/2021 12:17:31
I never thought it was any more complicated than this.
AFAIK it isn't - hence the "equivalent component" in engineering. There was a worrying case back in the 1970s of the rudders falling off RAF fighters. Problem was traced to a batch of tab washers manufactured in Czechoslovakia (as it then was) of equal dimensions but not equivalent fatigue tolerance.

Whilst this discovery has probably improved safety by tightening component certification, it has had some absurd effects: I enquired about replacing the four noisy, corroded headphone jacks on a school Cessna. Definitely not a life-critical, stressed, supportive or  aerodynamic part, so why not use an equivalent  2 jack socket from the guitar shop? Because any part fitted to a certified airplane must be certified equivalent, and the paperwork for part, electrical, passive, non-moving, costs 20 - each. So we put up with the scratchy reception. 

I digress, but this is a chat section after all. So a further digression, apropos "more lines → stronger". Police driving safety school brilliantly summarises road markings as "more paint → more danger". And they add that red paint costs more than white paint, so red paint → bloody dangerous. 

Weirdly, more lines can also mean less!  > does not allow equal, but ≥ does.
Title: Re: Equal or equivalent?
Post by: Bored chemist on 21/10/2021 12:53:33
Weirdly, more lines can also mean less!  > does not allow equal, but ≥ does.
Unless your  "! >" means "less than or equal to".
Confusing, isn't it.
Title: Re: Equal or equivalent?
Post by: alancalverd on 21/10/2021 14:23:24
It's not a sign I have seen in use. Specsavers time? Or have I missed a joke?
Title: Re: Equal or equivalent?
Post by: Bored chemist on 21/10/2021 18:44:26
It's not a sign I have seen in use.
Was your account hacked?

!  >
Or have I missed a joke?
Probably.
https://docs.microsoft.com/en-us/cpp/cpp/logical-negation-operator-exclpt?view=msvc-160#:~:text=The%20logical%20negation%20operator%20(%20!%20),to%20arithmetic%20or%20pointer%20type).&text=The%20result%20is%20true%20if,the%20converted%20operand%20is%20true%20.
Title: Re: Equal or equivalent?
Post by: chiralSPO on 21/10/2021 19:02:36
As far as I know, the symbol ≡ means identical to or congruent to. This is indeed generally the "strongest sameness" other than the tautological, "is itself".

Equals, as represented by = would be next, which is essentially the same as ≡ for simple things like numbers, but can diverge for more complex systems.

Equivalent is often the weakest of the three (while still being quite strong), and means "can be treated the same as" This is useful when making analogies (like isolobal or isoelectronic analogies in chemistry.)

However, when comparing "sameness" of components within a system, equivalent is stronger than identical, because it also indicates something about the environment. For example, three otherwise identical screws could be placed in non-equivalent positions within a device. When the device is assembled, these screws are still identical to each other, but are not equivalent to each other.
Title: Re: Equal or equivalent?
Post by: Bored chemist on 21/10/2021 19:07:23
In the interests of further confusion, chemists think " equivalent is a unit of quantity.
It's perfectly reasonable to ask you to weigh out an equivalent of zinc.

Title: Re: Equal or equivalent?
Post by: alancalverd on 21/10/2021 19:13:39
Quote from: alancalverd on Today at 14:23:24
Or have I missed a joke?
Probably.
https://docs.microsoft.com/en-us/cpp/cpp/logical-negation-operator-exclpt?view=msvc-160#:~:text=The%20logical%20negation%20operator%20(%20!%20),to%20arithmetic%20or%20pointer%20type).&text=
There's a difference between xxx! > and xxx !>, which Specsavers will be happy to point out. They are not equivalent. But it's a fair attempt at a very obscure joke, I'll grant you.
Title: Re: Equal or equivalent?
Post by: Eternal Student on 22/10/2021 14:04:35
Hi.

  Logical negation can also be written  with  bar  over the top of something.   a2dd007b2f7a53865f5bc6f55c5119ef.gif   means   Not(A).
So   ≡   obviously means  ≠     

Best Wishes.
Title: Re: Equal or equivalent?
Post by: alancalverd on 22/10/2021 14:23:24
Sheer brilliance, to reduce the slogans of NAACP and AAM to a mathematical inequality. Time for a T-shirt
                       ≡/≠
whose breathtaking ambiguity conveys the same message however you interpret it!