READING: I §3 9
Back to reading:
I §3 8
begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §3. FUNCTIONS. "end quote
These pages assume that the reader has the book and is reading along. topic index contents
Please read these notes from the beginning to here and then sentences 6, 7, and 8 of paragraph 7 of §3.
If f : X → Y is one-one onto then each element of Y corresponds under f to one and only one element of X, so we can define a function f : Y → X by f ( y ) = x if y = f(x). f is called the inverse function of f, and is also one-one onto.
Please note that the letter f and the supercript minus one together form the symbol which represents the inverse function.
Clearly ( f )= f, and f f = i, f f = i
The author of our text book was a man named Stephenson, and he wrote without proof that something was evident. Professor Bancroft came into class and said "What is evident to Stephenson is not evident to Bancroft." He proceeded to give a proof of Stephenson's evident fact. After the professor left, a classmate of mine said "And what is evident to Bancroft is not evident to the class." I hope I will have better luck than did Bancroft, because the three facts just quoted, which are clear to Rosenlicht, are not so clear to me. Let's look at the middle fact first:
f f = i
Let x and y be the elements of X and Y respectively which are connected by the function f. Recall that the composed function uses first the function that is to the right of the little circle, and then it uses the function on the left of the little circle. So this composed function just takes x into y and then y back into x. Consequently, we get the identity function on X, which was presented in the previous reading, I §3 8.
f f = i
Similarly, this just takes y into x and then x back into y, so we get the identity function on Y.
( f )= f
Here, the function inside the parentheses takes y into x. Then its inverse must take x back into y, and this is what the function f does.