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CSE115/ENGR160 Discrete Mathematics 02/17/

2.3 Functions

• Assign each element of a set to a particular element of a second set

Function and relation

• f:A→B can be defined in terms of a relation from A to B
• Recall a relation from A to B is just a subset of A x B
• A relation from A to B that contains one, and only one, ordered pair (a,b) for every element a ∈ A, defines a function f from A to B
• f(a)=b where (a,b) is the unique ordered pair in the relation

Domain and range

• If f is a function from A to B
  • A is the domain of f
  • B is the codomain of f
  • f(a)=b, b is the image of a and a is preimage of b
  • Range of f: set of all images of element of A
  • f maps A to B

Example

• G: function that assigns a grade to a student, e.g., G(Adams)=A
• Domain of G: {Adams, Chou, Goodfriend, Rodriguez, Stevens}
• Codomain of G: {A, B, C, D, F}
• Range of G is: {A, B, C, F}

Example

• Let R be the relation consisting of (Abdul, 22), (Brenda, 24), (Carla, 21), (Desire, 22), (Eddie,
  24. and (Felicia, 22)
• f: f(Abdul)=22, f(Brenda)=24, f(Carla)=21, f(Desire)=22, f(Eddie)=24, and f(Felicia)=
• Domain: {Abdul, Brenda, Carla, Desire, Eddie, Felicia}
• Codomain: set of positive integers
• Range: {21, 22, 24}

Example

• f: Z → Z, assigns the square of an integer to its integer, f(x)=x 2
• Domain: the set of all integers
• Codomain: set of all integers
• Range: all integers that are perfect squares, i.e., {0, 1, 4, 9, …}

Example

• In programming languages
  • int floor(float x){…} The domain of floor function is the set of real numbers and its codomain is the set of integers

Example

• f 1 (x) =x 2 and f 2 (x)= x-x 2
  • (f 1 +f 2 )(x)= f 1 (x) +f 2 (x)= x 2 + x-x 2 =x
  • (f 1 f 2 )(x)= f 1 (x) f 2 (x)= x 2 (x-x 2 )=x 3 -x 4

Function and subset

• When f is a function from A to B (f:A→B), the image of a subset of A can also be defined
• Let S be a subset of A, the image of S under function f is the subset of B that consists of the images of the elements of S
• Denote the image of S by f(S)
• f(S) denotes a set, not the value of function f 14

or{ ( )| }as shorthand

( ) { | ( ( ))} f s s S

f S t s S t f s ∈

= ∃ ∈ =

Example

• f maps {a,b,c,d} to {1,2,3,4,5} with f(a)=4, f(b)=5, f(c)=1, f(d)=
• Is f an one-to-one function?

Example

• Let f(x)=x 2 , from the set of integers to the set of integers. Is it one-to-one?
• f(1)=1, f(-1)=1, f(1)=f(-1) but 1≠-
• However, f(x)=x 2 is one-to-one for Z +
• Determine f(x)=x+1 from real numbers to itself is one-to-one or not
• It is one-to-one. To show this, note that x+1 ≠ y+1 when x≠y

Onto functions

• Onto : A function from A to B is onto or surjective , if and only if for every element b ∈ B there is an element a ∈ A with f(a)=b
• Every element of B is the image of some element in A

19

∀ y ∃ x ( f ( x ) = y ),where x isin thedomain and y is the codomain

f maps from {a, b, c, d} to {1, 2, 3}, is f onto?

Example

• Is f(x)=x 2 from the set of integers to the set of integers onto? - f(x)=-1?
• Is f(x)=x+1 from the set of integers to the set of integers onto? - It is onto, as for each integer y there is an integer x such that f(x)=y - To see this, f(x)=y iff x+1=y, which holds if and only if x=y-