1. CSC102 - Discrete Structures Functions

2. Definition of a function
A function takes an element from a set and maps it to a UNIQUE element in another set
f maps R to Z R Z
Domain
Co-domain f
f(4.3)
4.3 4
Pre-image of 4
Image of 4.3

3. A string length function
More functions
The image
of “a”
A pre-image
of 1
Domain
Co-domain A B C D F
Ali
Babar
Chishti
Dawood
Ammara
A class grade function 1 2 3 4 5
“a”
“bb“
“cccc”
“dd”
“e”
A string length function

4. Also not a valid function!
Even more functions
Range 1 2 3 4 5 a e i o u
Some function… 1 2 3 4 5
“a”
“bb“
“cccc”
“dd”
“e”
Not a valid function!
Also not a valid function!

5. Exercise
What can be domain and co-domain of a function that assigns the last two bits of a bit string of length 2?
4/26/2017

6. Function arithmetic

7. One-to-one functions
A function is one-to-one if each element in the co-domain has a unique pre-image
Formal definition: A function f is one-to-one if f(x) = f(y) implies x = y. 1 2 3 4 5 a e i o
A one-to-one function 1 2 3 4 5 a e i o
A function that is
not one-to-one

8. More on one-to-one Injective is synonymous with one-to-one
“A function is injective”
A function is an injection if it is one-to-one
Note that there can be un-used elements in a co-domain 1 2 3 4 5 a e i o
A one-to-one function

9. Exercise
4/26/2017

10. A function that is not onto
Onto functions
A function is onto if each element in the co-domain is an image of some pre-image
Formal definition: A function f is onto if for all y  C, there exists x  D such that f(x)=y. 1 2 3 4 a e i o u
An onto function 1 2 3 4 5 a e i o
A function that is not onto

11. More on onto Surjective is synonymous with onto
“A function is surjective”
A function is a surjection if it is onto
Note that there can be multiple used elements in the co-domain 1 2 3 4 a e i o u
An onto function

12. Exercise
4/26/2017

13. Onto vs. one-to-one
Are the following functions onto, one-to-one, both, or neither? 1 2 3 4 a b c 1 2 3 4 a b c d 1 2 3 4 a b c
1-to-1, not onto
Both 1-to-1 and onto
Not a valid function 1 2 3 a b c d 1 2 3 4 a b c d
Onto, not 1-to-1
Neither 1-to-1 nor onto

14. Bijections Consider a function that is both one-to-one and onto:1 2 3 4 a b c d
Consider a function that is both one-to-one and onto:
Such a function is a one-to-one correspondence, or a bijection

15. Identity functions
A function such that the image and the pre-image are ALWAYS equal
f(x) = 1*x
f(x) = x + 0
The domain and the co-domain must be the same set

16. Inverse functions If f(a) = b, then f-1(b) = a Let f(x) = 2*x R f R
4.3
8.6
f-1(8.6)
Then f-1(x) = x/2

17. More on inverse functions
Can we define the inverse of the following functions?
An inverse function can ONLY be done defined on a bijection 1 2 3 4 a b c 1 2 3 a b c d
What is f-1(2)?
Not onto!
What is f-1(2)?
Not 1-to-1!

18. Compositions of functions
(f ○ g)(x) = f(g(x))
f ○ g A B C g f
g(a)
f(b) a
f(g(a))
b = g(a)
(f ○ g)(a)

19. Compositions of functions
Let f(x) = 2x+3 Let g(x) = 3x+2
f ○ g R R R g f
g(1)
f(5)
f(g(1))=13 1
g(1)=5
(f ○ g)(1)
f(g(x)) = 2(3x+2)+3 = 6x+7

20. Compositions of functions
Does f(g(x)) = g(f(x))?
Let f(x) = 2x+3 Let g(x) = 3x+2
f(g(x)) = 2(3x+2)+3 = 6x+7
g(f(x)) = 3(2x+3)+2 = 6x+11
Function composition is not commutative!
Not equal!

21. Useful functions
Floor: x means take the greatest integer less than or equal to the number
Ceiling: x means take the lowest integer greater than or equal to the number
round(x) =  x+0.5 

22. Ceiling and floor properties
Let n be an integer (1a) x = n if and only if n ≤ x < n+1 (1b) x = n if and only if n-1 < x ≤ n (1c) x = n if and only if x-1 < n ≤ x (1d) x = n if and only if x ≤ n < x+1 (2) x-1 < x ≤ x ≤ x < x+1 (3a) -x = - x (3b) -x = - x (4a) x+n = x+n (4b) x+n = x+n

23. Ceiling property proof
Prove rule 4a: x+n = x+n
Where n is an integer
Will use rule 1a: x = n if and only if n ≤ x < n+1
Direct proof!
Let m = x
Thus, m ≤ x < m+1 (by rule 1a)
Add n to both sides: m+n ≤ x+n < m+n+1
By rule 1a, m+n = x+n
Since m = x, m+n also equals x+n
Thus, x+n = m+n = x+n

24. Factorial Factorial is denoted by n!
n! = n * (n-1) * (n-2) * … * 2 * 1
Thus, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Note that 0! is defined to equal 1

25. Proving Function problems
Let f be an invertible function from Y to Z
Let g be an invertible function from X to Y
Show that the inverse of f○g is:
(f○g)-1 = g-1 ○ f-1
(Pf) Thus, we want to show, for all zZ and xX ((f  g)  (g-1  f-1)) (x) = x and ((f-1  g-1)  (g  f)) (z) = z
((f  g)  (g-1  f-1)) (x) = (f  g) (g-1  f-1)) (x))
= (f  g) g-1 f-1(x)))
= (f g g-1 f-1(x)))))
= (f f-1(x)) = x
The second equality is similar
See Ex. 11, 12, 13, 14, 16, 17, 18, 19 on p138.
Q. 1, 3, 5, 7, 9, 10, 11, 12, 13, 15, 18, 19 ,27, 32, 33 on p-146