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Functions

- A function, f, is a mechanism that relates (or maps) one set of elements to another set.
- More specifically, f, is a special type of relation which associates the element of its domain to a unique element of its range.
- Let S and T be two sets, then f is often written as:
- f : S x T (f defined over a Cartesian Product just like relation)
- f : S T
- f (s) = t , for s in S (domain) and t in T (range)
- s f t , for s in S and t in T
- Formally, a function, f, over S x T is defined as follows:
- f = { \/ s: S; t1, t2: T I ( s ft1 /\ s f t2 ) -> t1 = t2 }

Function, f , pictorially

f (s1) = t1 and f (s2) = t1 ----- ok

f (s3) = t2 ----- ok

f (s4) = t3 and f (s4) = t4 ----- NOT ok

s1

t1

s2

t2

s3

t3

X

X

s4

t4

T = range of f

S = domain of f

Examples of functions

- f(x) = 2x + 2 , for x : N
- this is the same as y = 2x + 2 , for x, y : N
- g = { (1,2), (2,5), (3,10), ------ (n, n2+1)}, for n: N1
- Let S = { tom, jane, maple, sam} and Emp_N = N1, then f: S x Emp_N may be defined as:

f= { (tom, 3), (jane, 423) }

- Is Square Root a function?NO!

Sqrt (4) = +2 and Sqrt (4) = -2

Employee number is unique

This violates the definition

of a function because sqrt

results in two values.

Partial & Total Functions

- A function, f : S x T, is a partial function if dom f is a proper subset of S.
- dom f S (e.g. f(x) = 10/x does not include x = 0)
- A function, f: S x T, is a total function if the dom f is the same as S.
- dom f = S

Injection

- A function, f: S x T, is called an injection if
- f(s1) = t1 and f(s2) = t1 , then s1 = s2
- Injective functions are also called 1-to-1 functions

Note the inverse of f. If f is an injection, then f-1 is also a function

t1

s1

s2

t2

s3

t3

s4

X

s5

X

t4

f(s4) = t4 and f(s5) = t4

would not be allowed if

f were an injection

Examples of Injection

- f = {(1,3), (2,5), (3,2), (11,24) } is an injection
- Note that f-1 = {(3,1), (5,2), (2,3), (24,11)} is also a function.
- g = {(1,3), (2,5), (3,5), (11,24)} is NOT an injection
- Note that g-1 = {(3,1), (5,2), (5,3), (24,11)} is not a function with (5,2) and (5,3) as part of g-1
- So , when f is not an injection, f-1 will not be a function.
- Is “absolute value” function an injection?

I 4 I = 4 and I -4 I = 4

Surjection

- A function, f: S x T, is called a surjection if

ran f = T

- A surjective function is also called an ontofunction

S

T

t1

t2

t3

t4

t5

For function, f, to be surjective,

there can notbe t5 in T

Examples of Surjection

- Let A = { a1,a2,a3,a4,a5,a6,a7,a8,a9} and WK_day = { M,T,W,Th,F,S,Sn}, then g : A x WK_day defined below is a surjection

g = {(a1, T),(a2, M),(a3,Th),(a4, F),(a5, Sn),(a6,Sn), (a7, W), (a8,S)}

- but g-1 is not a surjection because a9 would not be included. (and what else can you say about g-1?)
- Is g-1 a total or partial function ? Is it even a function?
- Example: In a computing file system, the function, f, that maps file_owners to active_filesshould be a surjection because every active_file is owned by some file_owner. (But What do we have to look out for to make sure that f is even a function? May be it should be f-1? May be we should just leave “f” as a relation?)

Bijection

- A function, f, is called bijective if it both
- injective and
- surjective
- A bijective function is also known as isomorphic

Example of bijective function

- Let S = {0 and positive even integers} and T={positive odd integers}, then g: S x T defined below is a bijection.
- g (s) = t = s + 1 , for s in S and t in T
- note that g-1is also a bijection
- Let S = {positive integers} and then f : S x S defined below is NOT a bijection
- f(s) = s +1 , for s in S
- note that range of f does not include 1, which is in S.
- note also that the inverse function, f-1, can not include 1 as its domain because f-1(1) = 0 which is not in S.

Predecessor and Successor functions

- Let pred stands for predecessor function defined as:
- pred:N1 x N(note that N includes 0 and N1 does not)
- pred = {(1,0); (2,1); (3,2); - - - - -}
- Let succ stands for successor function defined as:
- Succ: N x N1
- Succ = { (0,1); (1,2); (2,3); - - - - - - }

Higher-Order functions

- A higher order function is a function, f, whose domain or range is itself a function.

Example of Higher-order function

- Model a query that will display all the items in a warehouse
- Let w = warehouse names = { Atl, Ny, SanFran, LA}, I = items = {shoes, boots, socks, pants, jackets}, and D = dozens of items = {0, 1, 2, 3}.
- Let function f : I x D be defined as the total function that specifies quantity of each item.
- Let g : w x P f be the higher-order function that specifies the amount of each item in the warehouses. {remember: P f stands for power set of f }
- g = { [Atl, ( (shoes,1),(boots,2),(socks,2),(pants,0),(jackets,1))], [Ny, ( (boots,2),(pants3),(jackets,1) ) ], - - - - }
- g (Atl) = {(shoes, 1), (boots, 2), (socks, 2), (pants, 0), (jackets,1) }

Higher-Order function example re-examine

- Look at the previous example:
- Would you want to redefine function f : I x D ?
- Would it be better to have a non-function, but just a relation for this model?

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