Maps, Functions, and Definition of One-to-One?

Question

By definition, a mapping or function states that each element in a set S has a unique element in the destination set, call it set T.

A mapping is one to one if f(x) = f(y) implies x = y.

My question then is, isn't every function, therefore, one-to-one?
If so, what is the point of defining the specific type of mapping known as one-to-one maps if, in order to be a mapping, this property must be satisfied.

yetterdn · Accepted Answer

No.  Your mistake is one of a common class of mistakes, namely confusing an implication and its converse.

In the definition of function x = y implies f(x) = f(y).  In the definition of one-to-one, f(x) = f(y) implies x = y, the converse of the first implication.

In general one implication can be true and its converse false.

In your specific case, consider an elementary function from the set of real numbers to the set of real numbers.  If f(x) = x^2, this is a function or mapping -- each real number has a unique square, and if x = y, x^2 = y^2.  It is not one-to-one, since, for instance
1^2 = (-1)^2, but 1 does not equal -1.

Jeremy Banta · Answer

No, because there exists some functions that are not one to one functions.there exists an infinite amount of counter examples one such such example is f(x)=x^2

this is not an one to one function because for example when x=2 the y value is 4, and when the x value is -2 then the y value is also 4. Because both different x-values are being mapped to the same particular y value creating an horizontal line. The function is by definition not an one-to-one function. There also exists numerous more cases like this one.

Another counter example is mapping all the vectors in R^3 to the Zero Vector in R^2 and this is not one-to-one everything in R^3 gets mapped to the zero vector in R^2. Is the Linear Algebra type of way of talking about it with transformations, and this would also be an linear transformation.

So the answer to your question is no..

chancebeaube · Answer

An easy example of a function that is NOTone-to-one is the line y=5.

Clearly it is a function but EVERY x maps to the same y.

Bob · Answer

Not every function is one-to-one. Consider the sine function: f(x) = sin(x)

sin(π/2) = sin(5π/2) = 1.

For the points x = π/2 and 5π/2, f(x) = f(y) but x ≠ y, so the sine function is not one-to-one.

However, the function f(x) = e^x is one-to-one since for all x, there is a unique value of f(x).

Elizabeth M · Answer

A function y=f(x) is one-to-one only if one value of x arises from one value of y in its range. In order to be a function there has to be only one value of y arising from one
value of x in the domain.
f(x)=x² in the domain x real defines a function but is not 1-1 since f(x)=4 for x=-2 and x=2.

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Maps, Functions, and Definition of One-to-One?

5 Answers