# “defined by” or “defined as”?

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

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]

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

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 ...”