1. Functions

2. Inverse Relation Let R be a relation from X to Y. R-1 = {yR-1x | xRy }
Examples of R & R-1:
x < y & y > x.
x | y & y is an integral multiple of x.

3. Functions R(a) = { b | aRb } is the image of a under R.
If R is an equivalence relation, then R(a) = [a].
Let A & B be nonempty sets.
A function f : A  B is a relation from A to B such that:
a  A,  b B, a f b, denoted f(a) = b
Every element of A has at least 1 image.
[ f(x) = y  f(x) = z ]  y = z.
Every element of A has at most 1 image.

4. Domain Let f : A  B. The domain of f is A. The co-domain of f is B.
The range of f is { b |  a  A, f(a) = b }.
Example: f : N  N, f(x) = 2x.
The co-domain of f is N.
The range of f is the even natural numbers.

5. Images
When f(a) = b, b is said to be the image of a under f (just as for general relations).
f-1(b) = { a | f(a) = b } is the set of preimages of b: the set of elements in A that map to b.
If f is an equivalence relation, is f-1(b) = [b]? Why?
For f : N  N, f(x) = 2x, what is f-1(3) ?

6. Visualizing Functions
Visualize functions via the vertical line test.
A relation that violates rule 1: every element has an image
A relation that violates rule 2: every element’s image is unique.
A graph that is discontinuous.
A graph where the co-domain  the range.

7. Surjective (onto) functions
Let f: X Y be a function.
f is surjective (aka onto) when the f’s range = f’s co-domain:
y Y, x X, f(x) = y.

8. Surjective (onto) functions ...
Examples:
f:   , f(n) = 2n is not surjective.
f:   , f(n) = 2n is surjective.
f:   , f(x) = x2 is not surjective.
f: Z  Z, f(x) = x - 21 is surjective.
f: Z  {0,1,2,3}, f(x) = x mod 4 is surjective.
f: +  + , f(x) = x2 is surjective.

9. Injective Functions Function f is injective when x  y  f(x)  f(y).
Examples:
f: Z  Z, f(x) = x2 (injective?)
f: Z  Z, f(x) = 2x (injective?)
f: Z  {0,1,2,3}, f(x) = x mod (injective?)

10. Bijective Functions
A function is bijective when it is surjective and injective.
A bijective function also is known as a 1-to-1 correspondence.
Examples:
f: Z  Z, f(x) = x (bijective?)
f:    , f(x) = 2x (bijective?)

11. Invertible Functions Let f: X  Y be a function.
Let f-1: Y  X be the inverse relation:
f-1 = {(y,x) | f(x) = y}.
Theorem: f-1 is a function if and only if f is a bijection.

12. Proof ( ) Proof ( ): If f-1 is a function then f is bijective.
y Y,  x X, f-1 (y) = x.
This means f is surjective. (Illustrate)
[f-1 (y) = x  f-1 (y) = z ]  x = z
This means f is injective. (Illustrate)
Therefore, f is bijective.

13. Proof ( ) Proof ( ): If f is bijective then f-1 is a function.
f is surjective: y Y,  x X, f (x) = y.
Equivalently, y Y,  x X, f-1 (y) = x. (Illustrate)
That is, every y has an image in X under f-1.
f is injective: x1  x2  f(x1)  f(x2).
Equivalently, f (x1) = f (x2)  x1 = x2.
Equivalently, (f-1 (y) = x1  f-1 (y) = x2)  x1 = x2.
That is, f-1 (y) is unique. (Illustrate)
Therefore, f-1 is a function.

14. Bijection Example Let f:   , f(x) = x2 When f is bijective,
f-1 = {(y,x) | x2 = y}
 x, both x & -x when squared produce x2.
Illustrate.
Transpose the vertical test to see if f-1 is a function.
When f is bijective,
f-1 ( f(x) ) = f-1 (y) = x
f ( f-1(y)) = f(x) = y

15. Composition of Functions
In general, functions do not commute:
Example:
r(x) = x + 1
s(y) = y2
Then,
s(r(x)) = (x + 1)2
r(s(x)) = x2 + 1

16. Characters
  
      
  
 
    
      