Functions, Inverse, and Relation

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 functionf : 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 4 (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 + 3 (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. • f-1 is a function: • 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. fis bijective: • 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 • 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 •    •        •    •   •      •       