CSE 20: Discrete Mathematics
for Computer Science
Prof. Shachar Lovett
2
Today’s Topics:
1.
2.
3.
Functions
Properties of functions
Inverse
3
1. Functions
4
Definition of a function
A
function 𝑓: 𝑋 → 𝑌 is a mapping that maps
each element of X to an element of Y
 Each
element of X is mapped to exactly
one element of Y
 Not two
 Not none
 Exactly one!
5
What is a function?
 Is
A.
B.
the following a function from X to Y?
Yes
No
X
Y
6
What is a function?
 Is
A.
B.
the following a function from X to Y?
Yes
No
X
Y
7
What is a function?
 Is
A.
B.
the following a function from X to Y?
Yes
No
X
Y
8
What is a function?
 Is
A.
B.
the following a function from X to Y?
Yes
No
Y
X
9
What is a function?
 Is
A.
B.
the following a function from Y to X?
Yes
No
Y
X
10
What is a function
A
function f:XY maps any element of X
to an element of Y
 Every
element of X is mapped – f(x) is
always defined
 Every
element of X is mapped to just one
value in Y
11
Properties of functions
Injective, surjective, bijective
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Injective, Surjective, Bijective…


Function f:X  Y
f is injective if: f(x)=f(y)  x=y


f is surjective if: yY xX s.t. f(x)=y



That is, no two elements in X are mapped to the
same value
There is always an “pre-image”
Could be more than one x!
f is bijective if it is both injective and surjective
13
Injective, Surjective, Bijective…
 Is
A.
B.
C.
D.
the following function
Injective
Surjective
Bijective
None
X
Y
14
Injective, Surjective, Bijective…
 Is
A.
B.
C.
D.
the following function
Injective
Surjective
Bijective
None
X
Y
15
Injective, Surjective, Bijective…
 Is
A.
B.
C.
D.
the following function
Injective
Surjective
Bijective
None
X
Y
16
Injective, Surjective, Bijective…
 Which
of the following functions f:NN is
not injective
A.
B.
C.
D.
E.
f(x)=x
f(x)=x2
f(x)=x+1
f(x)=2x
None/other/more than one
17
Injective, Surjective, Bijective…
 Which
of the following functions f:NN is
not surjective
A.
B.
C.
D.
E.
f(x)=x
f(x)=x2
f(x)=x+1
f(x)=2x
None/other/more than one
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Inverses
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Inverse functions


Functions 𝑓: 𝑋 → 𝑌 and 𝑔: 𝑌 → 𝑋 are inverses if
∀𝑥 ∈ 𝑋, 𝑔 𝑓 𝑥
=𝑥
∀𝑦 ∈ 𝑌, 𝑓 𝑔 𝑦
=𝑦
In this case we write 𝑔 = 𝑓 −1 (and also 𝑓 = 𝑔−1 )
20
Inverse functions
 Does
the following function have an
inverse:
f:R  R, f(x)=2x
A.
B.
Yes
No
21
Inverse functions
 Does
the following function have an
inverse:
f:Z  Z, f(x)=2x
A.
B.
Yes
No
22
Inverse functions
 Does
the following function have an
inverse:
f:{1,2}  {1,2,3,4}, f(x)=2x
A.
B.
Yes
No
23
Functions with an inverse are
surjective
 Let
f:XY, g:YX be inverse functions
 Theorem:
 Proof
f is surjective
(by contradiction):
 Assume not. That is, there is yY such that
for any xX, f(x)y.
 Let x’=g(y). Then, x’X and f(x’)=y.
 Contradiction. Hence, f is surjective. QED
24
Functions with an inverse are
injective

Let f:XY, g:YX be inverse functions

Theorem: f is injective

Proof (by contradiction):
Assume not. That is, there are distinct x1,x2X
such that f(x1)=f(x2).
Then g(f(x1))=g(f(x2)).
But since f,g are inverses, g(f(x1))=x1 and
g(f(x2))=x2.
So x1=x2.
Contradiction. Hence, f is injective. QED





25
Functions with an inverse are
bijective
 Let
f:XY, g:YX be inverse functions
 We
just showed that f must be both
surjective and injective
 Hence,
 It
bijective
turns out that the opposite is also true –
any bijective function has an inverse. We
might prove it later.