| Change log entry 45060 | |
|---|---|
| Processed by: | richwarm (2012-12-19 22:55:40 UTC) |
| Comment: |
<< review queue entry 43963 - submitted by 'alanwatson' >> 'one-to-one function' is not correct. The original 'one-to-one map between sets in math.' was ok, if not very precise, but your simpler version is misleading. For example a function can map a triplet so long as the set of all possible x,y,z and the set of values that they are mapped onto are the same. eg y=2x is an injection because y can range from -infinity to + infinity and so can x, but y=x squared (x real) s not because x can be negative but y is not, If you want to expand the English rather than just give the translation then you could put something like /injective map (ie a function in which the domain equals the image, math.)/ ============= Editor: Responding to six of your comments: 1) "your version is misleading" The two functions you take as examples are f: R --> R, f(x) = 2x g: R --> R, g(x) = x^2 Yes, f is an injection and g is not. But that is *equivalent* to saying ~ "f is a one-to-one function and g is not" It's simply a matter of standard nomenclature in mathematics. A "surjection" is the same thing as a "one-to-one function". "In mathematics, an injective function or an injection is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. ... An injective function is also said to be a one-to-one function." http://en.wikipedia.org/wiki/Injective_function 2) "injective map (ie a function in which the domain equals the image)" You seem to have a misconception about what an injective map is. It's not "a function in which the domain equals the image". There is no requirement of an injective function for the domain and image to have *anything* in common. Far from needing to be equal, they may be disjoint, in fact. e.g. f: {2, 3} --> {4, 6}, f(x) = 2x is a very simple function whose domain consists of two real numbers -- 2 and 3. Its codomain, which is also its image, is {4, 6}. f is an injective function, i.e. a one-to-one function (refer to Wikipedia's definition, quoted above). However, the domain and image are not equal. Indeed, they are disjoint: {2, 3} ∩ {4, 6} = ∅. 3) "y=2x is an injection because y can range from -infinity to + infinity and so can x" It's an injection, but your explanation of why is incorrect. That argument would apply equally well to the function f(x) = x^3 - x (x real), which is *not* an injection, since f(-1) = f(1). The reason "y=2x" is an injection is that distinct elements of the domain map to distinct elements of the codomain. In other words, 2x = 2y => x = y. 4) "... but y=x squared (x real) [is] not [an injection] because x can be negative but y [can] not" It's true that "y=x squared" is not an injection, but your explanation of why is incorrect. That argument would apply equally well to the function y=2^x (x real), which *is* an injection. The reason "y=x squared" is not an injection is that one can find distinct values x and y (e.g. x=3 and y=-3) such that x^2 = y^2. 5) "a function can map a triplet so long as the set of all possible x,y,z and the set of values that they are mapped onto are the same" I have no idea what you are talking about. 6) "The original 'one-to-one map between sets in math.' was ok" That's good to hear, because that definition was written by a professor of mathematics :-) |
| Diff: |
# - 單射 单射 [dan1 she4] /one-to-one function/injective map (math.)/ # + 單射 单射 [dan1 she4] /injective map (ie a function in which the domain equals the image math.)/ |