DISCRETE MATHEMATICSsri/Courses/DMATH/2020/DMATH2020...DISCRETE MATHEMATICS LECTURE 4-Reference: Plan: sections 8-5 and 8-b of-Principle of Inclusion-Exclusion book:-Applications of
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DISCRETE MATHEMATICS
LECTURE 4-
Reference :Plan :
sections 8-5 and 8 - b of
- Principle of Inclusion - Exclusion book:
- Applications of this principle DISCRETE MATHEMATICS
AND ITS APPLICATIONS- No - of onto functions
( 7th edition)- derangement by Kenneth Rosen
I Ai U Azl :n
t~"
*
'
,
-
IA , v Aal = IA , I + I Aal - I Ain Aal
↳ seen earlier as subtraction rule
I Ai U Aau Azl :
1
A Az
*¥ ox
As1 .
I t.
= IA , It IAZI + last - I Ain Aal - 1172nA> I - lash Aili r
p la T t IA , naan Asl
AE Ain Azn Ashby/ Ai U Az Vaz UA4 ) :
l l.
l l - 4
= 1 Ail t Hilt last tael
- b- IA , ? Az) - I Ain Az) - I Ain Aal
- - - - l Ash Aal
4. + I Ain Ash Azl + I Ain Asn Agl t 1 Azn Ash Ace )'
t 1 AT AznAal- IA, h Azn Az n Aa)
IA , v Az u - - - VAN l
why is this correct?
= £, I Ail -
every element at Awaz"-- uan
[ =p -
is counted exactly once- § Lain Aj)
A.
E Ai n Azn - . . Aj( Eicjsn III.JEN it;
1- 2, I Ain Agnate)a ¢ other Aa's .
is isjensn"d ' " i-tj.ae - tail - tj times
"
!I Ain 't;) - (t )
2
+ (- l )""
IA , naan . . - n An l thinkin And + ( Ig )I
←DJ't '
Contd . from previous slide :- -
In all intersections of more than jti Seti ,'a'
-is not counted .
- Expression giving the count of'a'is :
( I) - (E) +4 ) ... . . + rising ;)
- What is the value of this expression ?
- I Cvilhy ? )
( n +y )"= ¥; ( 7.) nnigi
N=l, y = - I
° = E:( 7) e. hi-- (on ) -(7) + ( z) - c ;) . . .
⇒ (7) - Indy ;) . ..
" Inn ) - I
IAI U Az U . . . VAN ) :
Principle of Inclusion - Exclusion :-- -
Let Ai , Az , . . . , An be finite sets
n
IAIUAZU - -- van ) = I taili=i
- §;I Ain Ail
+ E, I Ain Ajnttkli.jik
-
-- - a
+ I - I )""IA , n Azn . . . n Anl
Examine I : How many bit strings of length 8- centringfs?
PAUSE
-
Bit strings of lengths jbIHT:-/with 6 consecutive zeroes . x'T'
II
:-.
A , = { strings of length 8 that start with 6 consecutive zeroes }
Az = { strings - that have 6 consecutive zeroes
starting from position 23
As = {--
starting from position 33
Required answer = / A, U Aau As I = 4+4+4 - z- I -2+1In m - - -
IA , I = 4 LAI n Azl = 2 (Ain Azn Az) = 1
IAI = 4 I Ain As) it= 8
IA, I = 4 (Azn Az ) - 2
Number of onto functions :B-
1/1/1,set of all function
Yunnan.ms .
* in;;:÷.mg?mt9'' has
-
Ci = { functions where element i has no pre - image }
/ Ci Ucav - - - U Cn ) =Si Ici ) ( I ) In - 1)
m
- S. Icing. ) (nz ) Cn-2)m
+ E Icing- neap ( 3) In -37M'
.
M
i ( nm ,)(n- in - s))
:
+ C- l)"" I Goczn . - - n Cn ) O
Kil - in - in f- -
TajT
Icing ) = In -27M
(Cin cjnck) = ( n - 3)m
i'
.No . of onto functions :'
→
h-
"
nm .- fan) In - in + ( 2) In - 2)m - . . .
t C-is"
(n!) In - en-mm-
Derangement:-A permutation of objects so that no object is in its original position .
Examples : I 2 3-
-
t,
① 3 2.
2 I 13J2.3€ - de
3€-
tanganents .
3 ② I
Question : What is the number of derangement of a set-
with n elements ?
No . of derangement :#I 2 3 n - -
n
a::÷::.ftoriginal position derangement .
#
No - of permutations=n !
Ai = set of permutations where i is in its original position( ith )
IA , uh - -- u Ail = green part .
I Ail = ( n - i ) !
I Ain Ail = (n - z) !
:
#
IA, u Az . . want = ( Y ) ( n- i) !
- (2) In-4 !
+ (3) In -37!-
:
+ c-IT"
t !
# derangement = n ! - larvae "- - van)
No . of derangement of a set with n elements is :
Dna n ! - (7) Cn - D! + ( 2) In - z) ! - . . . + c- it (g) o!
non -negative integralExample : find the no. of solutions to n, t Mz t nz = 15- n
s -t . n , E 4 and nz E 8 and as E F
non -negative integralExample : find the no. of solutions to n, t Mz t as = 15- n
s -t . n , E 4 and nz E 8 and as E F-
" (l , 1,13 ) Solutions to Mit Nz -123=15- -
r
.
" " or:÷:
Nz E 8
Nz EF
Regd . answer = Total no. Of solutions
A , = set of solutions with Nc >4 - IA , uAzUA3 )
Az =- Nz > 8
Az = - n, > 7
NI t Mzt Nz = 15
Total no . Of solutions: 2 bars And It stars
= 17 Cz
# solutions with RI 34 . IAI )
n, = 95 -1N,') 5th! + nun , = is = 12 Czun
U,'t net n , = 10
IANMi + ni en> = 6 = 802
# solutions with me 78
IAS )# solutions with My> 7 M t na ng
'= 7 = 9 Cz
IA , n Azl . . .. - - (continue the calculation?
Summary:-
- Principle of Inclusion- Exclusion
- # of onto functions
-
# of derangement .
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