1. Functions Goals Introduce the concept of function
Introduce injective, surjective, & bijective functions

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Definition
Let D & C be nonempty sets.
A function f from D to C assigns elements of D to elements of C such that
For each d  D, f assigns d to exactly 1 element c  C, denoted f( d ) = c.
Equivalently
d  D c  C ( f( d ) = c  c’  C ( f( d ) = c’  c = c’ ) )
d  D !c  C f( d ) = c.
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If f is a function from D to C, we write f : D  C.
Functions also are known as:
Mappings - View f as a set of (key, value) pairs
{ (k, v) | k  D  v = f(k) }
Transformations.
Functions pass the vertical line test.
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Definition
A function is a subset of a Cartesian product: If f : D  C then f  D x C (see previous slide)
If f : D  C then:
D is f’s domain
C is f’s codomain.
If f( d ) = c then:
c is the image of d
d is a pre-image of c.
f’s range is { c  C | d f( d ) = c }.
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Example
Let f : Z  N be f( x ) = x2.
What is f’s domain?
What is f’s codomain?
What is the image of 4?
What is the pre-image of 4?
What is f’s range?
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6. Declaring a function’s domain & codomain
The Java statement
long square( int x ) { … }
The domain of square is?
Its codomain is?
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7. When are functions equal?
Let f1: D  C &
f2: D  C.
Since
A function is a subset of a Cartesian product.
A Cartesian product is a set.
when does f1 = f2 ?
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The Image of a Set
Let f : D  C and S  D.
The image of S under f, denoted f( S ) is
{ c  C | s  S, f( s ) = c }.
If S is finite, can | S | be <, =, or > | f( S ) | ?
Let f : N  N , f( n ) = n mod 5.
What is f’s range?
Let O = { n  N | n is odd } .
What is f( O ) ?
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9. One-to-One (Injective) Functions
Let f : D  C.
f is one-to-one (injective) when
a  D b  D ( a  b  f (a )  f ( b ) ).
Different domain elements have different images.
Example
Let n: { T, F }  { T, F }, such that n( p ) =  p.
Is n injective?
Is f : Z  Z, f( z ) = z2 injective?
Injective functions pass the horizontal line test.
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10. Onto (Surjective) Functions
Let f : D  C.
f is onto (surjective) when
c  C d  D ( f ( d ) = c ).
f’s range equals its codomain.
Example
Let or : { T, F }  { T, F }  { T, F }, such that or( p, q ) = p  q.
Is or surjective?
Is f : Z  Z, f( z ) = z2 surjective?
Is f : Z  Z, f( z ) = z mod 5 surjective?
Is f : Z  { 0, 1, 2, 3, 4}, f( z ) = z mod 5 surjective?
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11. One-to-One Correspondence (Bijection)
Function f is a one-to-one correspondence (bijection) when it is both:
one-to-one (injective)
onto (surjective).
Let f : R  R, f( x ) = 2x – 7. Is f a bijection?
Let f : D  C be a bijection, where D, C are finite.
Can |D| > |C|?
Can |D| < |C|?
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Inverse Functions
Let g : D  C be a bijection.
The inverse function of g, denoted g-1, is the map: C  D such that if g( d ) = c, then g-1( c ) = d.
If g is bijective, g-1 is a function because g is:
onto: c  C ( c is the image of some element in D )
1-to-1: c  C (c is the image of at most 1 element in D )
Diagram this.
If g : D  C is not a bijection, does g-1 exist (as a function)?
Always? Sometimes? Never?
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13. Composition of Functions
Let functions f : B  C & g: A  B.
The composition of f & g, denoted f  g, is defined as f  g( a ) = f( g( a ) ), for a  A a
f(g( a ))
g( a ) A B C g f
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Example
Let f : Q  Q, f( x ) = 2x + 1.
Let g : Q  Q, g( x ) = ( x – 1 ) / 2.
What is ( g  f )( 17 )?
In general, what is g-1  g ( x ) ?
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Exercise
Let S  U. The characteristic function of S
fS : U  { 0, 1 } is such that
x  S  fS ( x ) = 1
x  S  fS ( x ) = 0.
Show that:
f A  B (x) = fA( x )fB( x )
f A  B (x) = fA( x ) + fB( x ) - f A  B ( x ) A B 3 4 2 1
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End of Lecture
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The Java statement
long square( int x ) { … }
square’s domain is int; its codomain is long.
Let f & g be functions from A to R.
(f + g)( x ) = f( x ) + g( x ),
( fg )( x ) = f( x )g( x ).
Let f( x ) = x2 and g( x ) = x – x2.
What is f + g? gf?
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Graphs of Functions
Let f : A  B.
The graph of f = { (a, b) | a  A and f( a ) = b }.
Example: Let the domain of f be N. Draw:
f( x ) = x2
f( x ) = x mod 2.
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