Extensible Networking Platform 11 - CSE 240 – Logic and Discrete Mathematics
Announcements
• Homework 5 was due this morning
• Homework 6 is posted online
• Read Section 2.4 (Sequences and Summations) by Wednesday
• Quiz 4 is today
• Quiz 5 is on Wednesday
Extensible Networking Platform 22 - CSE 240 – Logic and Discrete Mathematics
Functions
Suppose we have:
And I ask you to describe the yellow function.
Notation: f: R®R, f(x) = -(1/2)x - 25
What’s a function? f(x) = -(1/2)x - 25
domain co-domain
Extensible Networking Platform 33 - CSE 240 – Logic and Discrete Mathematics
Functions
Definition: a function f : A ® B is a subset of AxB where " a Î A, $! b Î B and <a,b> Î f.
A
B
A
B
A point!
A collection of points!
Extensible Networking Platform 44 - CSE 240 – Logic and Discrete Mathematics
Functions
A = {Michael, Tito, Janet, Cindy, Bobby}B = {Katherine Jackson, Carol Brady, Mother Teresa}
Let f: A ® B be defined as f(a) = mother(a).
Michael Tito
Janet Cindy
Bobby
Katherine Jackson
Carol Brady
Mother Teresa
Extensible Networking Platform 55 - CSE 240 – Logic and Discrete Mathematics
Functions - image & preimage
For any set S Í A, image(S) = {b : $a Î S, f(a) = b}
So, image({Michael, Tito}) = ?and image(A) = ?
What about the range?
Range is all values the
function maps to.
image(S) = f(S)
Michael
Tito
Janet
Cindy
Bobby
Katherine Jackson
Carol Brady
Mother TeresaWhat about
the codomain?
Everything in B
{Katherine Jackson}
B - {Mother Teresa}
Extensible Networking Platform 66 - CSE 240 – Logic and Discrete Mathematics
Functions - image & preimage
For any S Í B, preimage(S) = {a: $b Î S, f(a) = b}
So, preimage({Carol Brady}) = ? preimage(B) = ?
preimage(S) = f-1(S)
Michael Tito
Janet Cindy
Bobby
Katherine Jackson
Carol Brady
Mother Teresa
{Cindy, Bobby}A
Extensible Networking Platform 77 - CSE 240 – Logic and Discrete Mathematics
Functions - injection
A function f: A ® B is one-to-one (injective, an injection) if "a,b,c, (f(a) = b Ù f(c) = b) ® a = c
Not one-to-one
Every b Î B has at most 1 preimage.
Michael Tito
Janet Cindy
Bobby
Katherine Jackson
Carol Brady
Mother Teresa
Extensible Networking Platform 88 - CSE 240 – Logic and Discrete Mathematics
Functions - surjection
A function f: A ® B is onto (surjective, a surjection) if "b Î B, $a Î A f(a) = b
Not onto
Every b Î B has at least 1 preimage.
Michael Tito
Janet Cindy
Bobby
Katherine Jackson
Carol Brady
Mother Teresa
Extensible Networking Platform 99 - CSE 240 – Logic and Discrete Mathematics
Functions - bijectionA function f: A ® B is bijective if it is one-to-one
and onto (also called a one-to-one correspondence).
Isaak Bri Lynette
Aidan Evan
Cinda Dee Deb Katrina Dawn
Every b Î B has exactly 1 preimage.
An important implication of this
characteristic:The preimage (f-1) is a
function!
Isaak Bri Lynette
Aidan Evan
Cinda Dee Deb Katrina Dawn
Extensible Networking Platform 1010 - CSE 240 – Logic and Discrete Mathematics
Functions - examples
Suppose f: R+ ® R+, f(x) = x2.
Is f one-to-one?Is f onto?Is f bijective?
yes
yes
yes
Extensible Networking Platform 1111 - CSE 240 – Logic and Discrete Mathematics
Functions - examples
Suppose f: R ® R+, f(x) = x2.
Is f one-to-one?Is f onto?Is f bijective?
yes
no
no
Extensible Networking Platform 1212 - CSE 240 – Logic and Discrete Mathematics
Functions - examples
Suppose f: R ® R, f(x) = x2.
Is f one-to-one?Is f onto?Is f bijective?
no
no
no
Extensible Networking Platform 1313 - CSE 240 – Logic and Discrete Mathematics
More Examples
Extensible Networking Platform 1414 - CSE 240 – Logic and Discrete Mathematics
Functions - composition
Let f:A®B, and g:B®C be functions. Then the composition of f and g is:
(g o f)(x) = g(f(x))
Extensible Networking Platform 1515 - CSE 240 – Logic and Discrete Mathematics
Functions - compositionLet f1 and f2 be functions from the set of integers to the set
of integers defined by f(x) = 2x+3 and g(x) = 3x+2
(f o g)(x) = f(g(x)) = f(3x+2) = 2(3x +2) +3 = 6x + 7
(g o f) (x) = g(f(x)) = g(2x + 3) = 3(2x+3) + 2 = 6x + 11
Extensible Networking Platform 1616 - CSE 240 – Logic and Discrete Mathematics
Sequences
Definition:A sequence {ai} is a function f: N È {0} ® R, where we write ai to
indicate f(i).
Examples:
Sequence {ai}, where ai = i is just a0 = 0, a1 = 1, a2 = 2, …
Sequence {ai}, where ai = i2 is just a0 = 0, a1 = 1, a2 = 4, …
Extensible Networking Platform 1717 - CSE 240 – Logic and Discrete Mathematics
Summation
The symbol:
Lower limiti=1
Upper limiti=k
€
aii=1
k
∑ = a1 + a2 + … + ak
Extensible Networking Platform 1818 - CSE 240 – Logic and Discrete Mathematics
Examples
What is the value of ?
= 12 + 22 + 32 + 42 + 52
= 1 + 4 + 9 + 16 + 25 = 55
What if we want the index to run from 0 – 4 instead of 1 – 5?
=1 + 4 + 9 + 16 + 25 = 55
å=
5
1
2
ii
åå==
+=4
0
25
1
2 )1(jiji
Extensible Networking Platform 1919 - CSE 240 – Logic and Discrete Mathematics
Summation
How do you know this is true?
€
cai + bi( )i=1
k
∑ = c aii=1
k
∑ + bii=1
k
∑
• Use associativity to separate the b from the a.
• Use distributivity to factor the c.
Extensible Networking Platform 2020 - CSE 240 – Logic and Discrete Mathematics
Summations you should know…
What is S = 1 + 2 + 3 + … + n?
• You get n copies of (n+1). But we�ve over added by a factor of 2. So just divide by 2.
S = 1 + 2 + … + n
S = n + n-1 + … + 1
2s = n+1 + n+1 + … + n+1
Write the sum.
Write it again.
Add together.
€
kk=1
n
∑ =n(n + 1)2
Extensible Networking Platform 2121 - CSE 240 – Logic and Discrete Mathematics
Summations you should know…
What is S = 1 + 3 + 5 + … + (2n - 1)?Sum of first n odds.
€
(2k −1)k=1
n
∑ = 2 kk=1
n
∑ − 1k=1
n
∑
€
= 2 n(n + 1)2
"
# $
%
& ' − n
€
= n2
Extensible Networking Platform 2222 - CSE 240 – Logic and Discrete Mathematics
Summations you should know…
What is S = 1 + 3 + 5 + … + (2n - 1)?
Sum of first n odds.
€
= n2
Extensible Networking Platform 2323 - CSE 240 – Logic and Discrete Mathematics
Infinite Cardinality
Two sets A and B have the same cardinality if and only if there exists a bijection (one-to-one correspondence) between them, A ~ B.
An infinite set is �countably infinite� if it can be put into �one-to-
one correspondence (bijection) � with the set
of natural numbers.
A set is “countable” if it is either finite or countably infinite.
Extensible Networking Platform 2424 - CSE 240 – Logic and Discrete Mathematics
Infinite Cardinality
• If there exists a function f from A to B that is injective (i.e. one-to-one) we say that |A| |B|
• If there exists a function f from A to B that is surjective (i.e. onto) we say that |A| |B|
£
³Why?
Why?
Extensible Networking Platform 2525 - CSE 240 – Logic and Discrete Mathematics
Set of Java programs countably infinite?
• How would I count them?
• What if I created a Java program generator– Program 1 = �a�– Program 2 = �b�– Program 3 = �c�....
• Eventually you would find programs that compile
• What if I count these programs?
Extensible Networking Platform 2626 - CSE 240 – Logic and Discrete Mathematics
Are rational numbers countable?
• How would I count them?
• What method would I use to list all of the rational numbers?
Extensible Networking Platform 2727 - CSE 240 – Logic and Discrete Mathematics
Infinite Cardinality
Are there more evens than odds?
Are there more natural numbers than evens?
Are there more evens than multiples of 3?
{0,2,4,6,8,…} ~ {1,3,5,7,9,…}, f(x) = x-1
N ~ {0,2,4,6,8,…}, f(x) = 2x
{0,2,4,6,8,…} ~ {0,3,6,9,12,…}, f(x) = 3x/2
Extensible Networking Platform 2828 - CSE 240 – Logic and Discrete Mathematics
Infinite Cardinality
How many rational numbers are there?
1/1, 1/2, 1/3, 1/4, …2/1, 2/2, 2/3, 2/4, …3/1, 3/2, 3/3, 3/4, …
…
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, …
Extensible Networking Platform 2929 - CSE 240 – Logic and Discrete Mathematics
Infinite Cardinality
How many real numbers are in interval [0, 1]?
More irrational between 0 and 1 then all rationaleverywhere
0.4 3 2 9 0 1 3 2 9 8 4 2 0 3 9 …0.8 2 5 9 9 1 3 2 7 2 5 8 9 2 5 …0.9 2 5 3 9 1 5 9 7 4 5 0 6 2 1 …
…
�Countably many! There�s the list!�
�Are you sure they’re all there?�
Counterexample:0.5 3 6 …
So we say the reals are �uncountable.�
Extensible Networking Platform 3030 - CSE 240 – Logic and Discrete Mathematics
Quiz 4