There are a great many functions that fail the horizontal line test (HLT) which nevertheless have an inverse function.
How do we resolve this?
We know that the inverse of sin function exists, it is right on the calculator, which clearly contradicts with my statement earlier. This contradiction can be resolved using the notion of restriction.
The graph of a sine function is given,
If one considers the whole function, it obviously fails the HLT. However, if we restrict the domain of the sine function to the closed interval,
A winding map is a function that goes from
The output of this function is unusual in that the output were ordered pairs. Basically, we have an ordered pair, in which the second element is an ordered pair. Another way of expressing a winding map:
There are other interesting functions we will encounter, all of which are aspects of the characteristic functions of a subset.
Consider a codomain,
Another interesting function is the Iverson functions, uses a shorthand
Iversion brackets are particularly useful in expressing and simplifying sums. We can write the number of all even numbers less than 25 as,
For this example:
Kronecker delta can be considered as a special case of the idea inherent in Iverson bracket. The function is written as
The Iverson bracket equivalent of Kronecker delta is
Disclaimer: this is a summary of section 6.6 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, most of the equations are screenshots of the book and the same examples are treated.
