Difference between "Identical", "Equal", "Equivalent"
as topic, thanks
In what context?
hmm, in mathematics....right?
firstly, is there any difference between identical and equal?
When we are talking about congruent triangles,
ABC is identical to EFG
can someone say, ABC is equal to EFG?
or can we say, ABC is the same as EFG?
how do we determine which one we have to use?
I've never heard anyone say "ABC is identical to EFG" in the case of triangles. Congruent, yes, identical, no. But I think if you are considering ABC identical to EFG for whatever purposes, then it is also probably reasonable to say they are equal or the same as each other, speaking informally.
I've never seen "identical" used in a mathematical context except as a synonym for "congruent mod" some number. "Equal" means "are names for the same thing". That is 5*3 equals 15 because they are just different ways of talking about the same number. If I say "triangle ABC" is equal to "triangle XYZ", I am saying something quite different for "congruent"- I am saying that those are just different ways of referring to exactly the same triangle.
"Equivalent" means "the same in some specific way" and can be used in a variety of ways.(Essentially the idea of an equivalence relation.)
I think Orthodontist might disagree with you HallsofIvy, if you saw our posts on the processor problem.
I think it's more of a qualitative term that math text authors use.
edit: 'identical' that is
Very very loosely, equals is used to indicate that two things are 'uniquely' the same, that is there is all choices required to specify each one are the same. In this sense we have 4=2^2. There are no choices: 2 is 2, 4 is 4 and if you square 2 you get 4.
Now, the vector space R^2, the x-y plane is absolutely not equal to the set of polynomials R[x] modulo x^2. One is a polynomial ring modulo an ideal, and the other just *isn't*. Yet, as vector spaces they are equivalent (or isomorphic).
Equivalence is often used these days in regards to Category Theory. Categories of objects (and maps) are often equivalent yet in no sense are they equal (one will often contain a set of objects, and the other a proper class).
You should, perhaps, think of 'equivalence' as being a qualified statement: up to some characteristics these things are indisinguishable in some sense (a sense that can be made precise, but changes depending upon the situation), yet equals means: these things are absolutely and unequivocally the same object.
Equal is used when 2 equations are numerically equal in some cases.
eg if I say x^2 = (equal) x, I am looking for a number(s) which when replaced with x, both sides give the same number.
Identical is used when 2 equations have the same effect but written in a different way.
eg y=x and y=x+1-1 are identical, one can be modified into the other. Basically you know you have 2 identical equations when both give exactly the same outputs when given the same inputs or when both give the same graph when plotted.
As for equivalent, I don't think it fits into my context of algebra...
Re: Difference between "Identical", "Equal", "Equivalent"
If you have one-egged identical twins: they are not identical for instance they should have different social security number. So identification I see in set theory: you have the set of objects in your sachel. It doesn't contain machinely manifacturered new replications more than one time. So all objects are different. This in opposition to identical. Sets can contain things which don't even have the same unity to describe them with. If these units are the same, just then there is the possibility of equal: Two things qua identity different but with a property that is equal.
For instance the identical twins have the same shirt-size. It could be that they have different shoe-size or have excatly the same feet but one likes it shoes too sit loose and the other likes them fit.
Ok the difference between equal and identical already clear?
Equivalent has something to do with geometrical equality. In meteorology (perhaps also elsewere) one speaks of dimensionless numbers. A MODEL IS TOTALLY DIFINED BY A NUMBER FOR ALL OF IT DIMENSIONLESS NUMBERS. So in this definition all squares are equal, yet in many cases (all but one) non-identical. Yet:
I don't think that two models are equivalent if the values for their dimensionless numbers are equal, but when the relation between the dimensionless numbers are alike!
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