Functions

1. f( ) = A B Functions Lecture 12

2. Functions function, f, from set A to set B associates an element , with an element The domain of f is A. The codomain of f is B. For every input there is exactly one output.

3. Functions

4. Functions f(S) = |S| f(string) = length(string) f(student) = student-ID f(x) = is-prime(x)

5. f( ) = A B Injections (one-to-one) is an injection iff every element of B is f of at most 1 thing ≤ 1 arrow in |A| ≤|B|

6. f( ) = Surjections (Onto) is asurjectioniff every element of B is f of something 1 arrow in A B |A| ≥|B|

7. f( ) = A B Bijections is a bijection iff it is surjection and injection. exactly one arrow in |A| =|B|

8. Functions a. One-to-one, b. Onto, c. One-to-one, d. neither d. Not a Not onto not one-to-one and onto function a 1 a a 1 a 1 1 b 2 b 1 b 2 b 2 a 2 c 3 c 2 c 3 c 3 b 3 4 d 3 d 4 d 4 c 4 8

9. In-Class Exercises

10. Inverse Sets A B Given an element y in B, the inverse set of y := f-1(y) = {x in A | f(x) = y}.

11. f( ) = A B Inverse Function Informally, an inverse function f-1 is to “undo” the operation of function f. exactly one arrow in There is an inverse function f-1 for f if and only if f is a bijection.

12. Composition of Functions Two functions f:X->Y’, g:Y->Z so that Y’ is a subset of Y, then the composition of f and g is the function g。f: X->Z, where g。f(x) = g(f(x)). Y’ Z X Y

13. In-Class Exercises

14. In-Class Exercises