8

MY main interest is in the mathematical context, where one defines objects by a formula. I can imagine 2 cases :

  1. Direct case, "direct definition": The function f is defined by/as f=a+b+c
  2. Implicit definition: The number x is defined by/as a = g(x)

I would tend to use as for the case 1 and by for the second one.

  • If pressed, I'd use by in the second case (the function has to be 1 to 1 for x to be defined, of course) - but why not then write x = g [index -1] (a) ? (Sorry, I can't use the normal superscript characters to show the inverse function.) I'd use thus (or a colon followed on the next line by the formula) in the first case. – Edwin Ashworth Jul 11 '13 at 8:08
  • @EdwinAshworth x = g⁻¹(a) -- Unicode has a large number of characters available via charmap in Windows. – Andrew Leach Jul 11 '13 at 8:48
5

I would agree with the use of "by" in both your examples, but if I were writing something like the following examples I would use "as":

The value of x is defined as equal to the sum a + b
In this example, x is defined as equivalent to ...

So:

... defined by [a formula]
... defined as [a value]

0

For the case 1, I think 'by' is more correct because 'by ...' is an abbreviation for 'by the formula ...'.

0

Grammatically, both by and as are acceptable in either of the examples given. But as mathematical statements, both examples have problems.

In the first example, “The function f is defined by/as f=a+b+c”, unless a, b, c all are previously-defined functions or constants, the arguments of f and its dependence on them is unclear, ie undefined, hence the statement that “f is defined” is improper. Instead say (for example) “Let f(x) = a+b+c(x)” [in the case where a, b are constants and c(x) is a function].

In the second example, as is likely to be unacceptable to many mathematicians, and by, while acceptable, misleading. That is, the expression “x is defined as a = g(x)” would be viewed as nonsense, and “x is defined by a = g(x)” as an assertion that needs to be proved, rather than as a definition, because g⁻¹(a) might be undefined or might have multiple values. Instead say “Let x be such that a = g(x). To see that x is well-defined, consider ...”

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