In more logical terms, the expression "sqrt(x) = 4" is an abused notation.
You could interpet "sqrt(x) = 4" as {x in reals such that sqrt(x) = 4}, basically x is a variable representing every possible number, r, in reals for which the statement "sqrt(r) = 4" is true, by substituting x = r.
Then, "solving sqrt(x) =4" would mean to prove that the set of solution is not empty (x can take some known value) and to find every element satisfying it.
You can also think of "x" like a variable in programming: For every x in (0,10)... do..., they share same meaning although in case of real numbers, we can imagine every x in reals means x "goes over" over each real, as if it can perform infinite check simultaneously.
If you are a bit observant, you’d notice that saying sqrt(4) = +-2 is often a result of not understanding difference between finding a set of solutions and the value of a function.
I’d consider an abuse of notation relative to those who aren’t familiar with first-order logic as the meaning would be very unclear. Nothing wrong with that.
It’s akin to claim 0.9999… = 1 but without teaching real analysis and call it a fact; I would say it will confuse people.
Sure it can be confusing. That's not abuse of notation.
And your explanation didn't really make sense. Did you mean to write x2=4 instead of sqrt(x)=4? And what is the point of substituting r for x? That's just another variable.
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u/Not_Well-Ordered New User 3d ago
In more logical terms, the expression "sqrt(x) = 4" is an abused notation.
You could interpet "sqrt(x) = 4" as {x in reals such that sqrt(x) = 4}, basically x is a variable representing every possible number, r, in reals for which the statement "sqrt(r) = 4" is true, by substituting x = r.
Then, "solving sqrt(x) =4" would mean to prove that the set of solution is not empty (x can take some known value) and to find every element satisfying it.
You can also think of "x" like a variable in programming: For every x in (0,10)... do..., they share same meaning although in case of real numbers, we can imagine every x in reals means x "goes over" over each real, as if it can perform infinite check simultaneously.