exámenes de matemática discreta

7/26/2019 Exmenes de Matemtica Discreta

1/105


2/105


3/105


4/105

12 + 22 + 32 + ... + (n 1)2 + n2 =n(n+ 1)(2n+ 1)

6

f : A B

A

f

f(a) =f(b)

a= b

f

A

B

a

b

c

a

c


5/105

12 + 22 + 32 + ... + (n 1)2 + n2 =n(n+ 1)(2n+ 1)

6

n= 1

12 = 1 =1 2 3

6

n

n + 1

12 + ... + n2 + (n+ 1)2 =(n + 1)(n+ 2)(2n+ 3)

6

12 + ... + n2 + (n+ 1)2 = n(n+ 1)(2n+ 1)6

+ (n+ 1)2.

(n+ 1)(n+ 2)(2n+ 3)

6 .

a1,...,an

s:= 0

i= 1

n

s= 0

j =i + 1

n

s= 0

ai=aj

s:= 1

s= 0

r:=

r:=

r


6/105

O(n2)

i

n

j

n i

f : A B

A

f

f(a) =f(b)

a= b

f

A

B

A

n

A

1, 2,...,n

B

1, 2,...,m

m

B

f(1), f(2),...,f(n)

f

A

a

b

c

a

c

a

b

b

a= (a0, a1,...,an)b, b , cp:= a0, i:= 0

(i < n)i:= i + 1p:= p + aib

i

p

b


7/105

x 3(mod5)x 5(mod7)

x 9(mod11).

A

B

A

B

210

2210

22210

52 283

5 4 28 27 26

5 4 283

x

y

xayb

a

b

a+ b = n

18 17

9 17


8/105

182

5!

5!/4

5!/5

(3 2 36)/(39 38 37)

(4 3 2 36)/(39 38 37)

(3 3 2 36)/(39 38 37)

n

p

n

3

pn3(1 p)3

3i=0n

i

pni(1 p)i

A

X

A

E(X)

E

F

E

F


9/105

F

E

a

b

mcd(a, b) = 1

mcd(a + b, a b) = 1

mcd(a + b, a b) = 3

mcd(a + b, a b) = 1

mcd(a + b, a b) = 2

mcd(a + b, a b) = 21

mcd(a + b, a b) = 3

mcd(234, 567)


10/105

3 5 = 15

x 3(mod5)x 5(mod7)

x 9(mod11).

5 7 11 = 385

A

B

A

B

210

2210

22210

n

2n

52 283


11/105

5 4 28 27 26

5 4 283

x

y

n

xayb

a

b

a+ b = n

18 17

9 17

18

a

b

b= 17 a

a

a

0

17

5!

5!/4

5!/5

5!

(3 2 36)/(39 38 37) (4 3 2 36)/(39 38 37)

(3 3 2 36)/(39 38 37)

3/39


12/105

2/38

36/38

(3 2 36)/(39 38 37).

n

p

n

3pn3(1 p)3

3i=0nip

ni

(1 p)i

A

X

A

E(X)

p1,...,pn

p1 ... pn + 1

pi


13/105

103 2

E

F

E

F

F

E

a

b

a > b

mcd(a, b) = 1

mcd(a + b, a b) = 1

mcd(a + b, a b) = 3

mcd(a + b, a b) = 1

mcd(a + b, a b) = 2

mcd(a + b, a b) = 21

mcd(a + b, a b) = 3

p= mcd(a + b, a b)

a + b= ps

a b= pt

2a= p(s + t)

2b= p(s t).

p

2a

2b

p

mcd(2a, 2b)

p= 2

p

a

b

p= 1


14/105

mcd(234, 567)


15/105

x 2(mod5)x 4(mod7)

x 8(mod11).

(3 2 36)/(39 38 37)

(4 3 2 36)/(39 38 37)

(3 3 2 36)/(39 38 37)

x

y

xayb

a

b

a+ b = n

18 17

9 17


16/105

182

52 282

5 4 28 27

5 4 282

6!/4

5! 6!/5

n

p

n

3

pn3(1 p)3

3i=0n

i

pni(1 p)i

B

X

A

E(X)

a

b

mcd(a, b) = 1

mcd(a + b, a b) = 1

mcd(a + b, a b) = 3

mcd(a + b, a b) = 1

mcd(a + b, a b) = 2

mcd(a + b, a b) = 21

mcd(a + b, a b) = 3


17/105

mcd(234, 567)

A

B

A

B

27

2227

227

E

F

E

F

F

E


18/105

4 2 = 8

x 2(mod5)x 4(mod7)

x 8(mod11).

5 7 11 = 385

(3 2 36)/(39 38 37)

(4 3 2 36)/(39 38 37)

(3 3 2 36)/(39 38 37)

3/39

2/38


19/105

36/38

(3 2 36)/(39 38 37).

p1,...,pn

p1 ... pn + 1

pi

x

y

xayb

a

b

a+ b = n

18 17

9 17

18

a

b

b= 17 a

a

a

0

17

52 282

5 4 28 27

5 4 282


20/105


21/105

p= mcd(a + b, a b)

a + b= ps

a b= pt

2a= p(s + t)

2b= p(s t).

p

2a

2b

p

mcd(2a, 2b)

p= 2

p

a

b

p= 1

mcd(234, 567)

A

B

A

B

27

2227

227

n

2n


22/105

E

F

E

F

F

E


23/105

a R

a > 0

n

(1 + a)n 1 + an.

((p q) (q r)) (p r).

f : Z7 Z7

f(a) =5a+ 1

Z7

f

g: Z7 Z7

g f

(g f)(a) =g(f(a))

x3 mod 5

x5 mod 7

x2 mod 2

a1, a2,...,an

a

1 b n +1

a

b


24/105

a R

a > 0

n

(1 + a)n 1 + an.

n= 1

1 + a 1 + a.

n

n + 1

(1 + a)n+1 = (1 + a)n(1 + a) (1 + an)(1 + a)

(1 + an)(1 + a) = 1 + a + an+ a2n.

1 + a + an+ a2n 1 + a(n + 1) = 1 + an + a.

a2n 0,

a >0

((p q) (q r)) (p r).

p:= F

r:= V

q:= V

(p q) :=V

(q r) :=V

((p q) (q r)) :=V .


25/105

(p r) :=F

V F

U

A

U A

P((x, y)) :=el alumno x ha aprobado la asignatura y.

xUy A (P(x, y)).

x Uy A P(x, y).

f : Z7 Z7

f(a) =5a + 1

Z7

f

f(0) = 1f(1) = 6f(2) = 4f(3) = 2f(4) = 0f(5) = 5f(6) = 3

f

g: Z7 Z7

g f

(g f)(a) =g(f(a))

f

g

g(b) = 3b + 4


26/105

g(f(a)) =g(5a + 1) = 3(5a+ 1) + 4 =a + 3 + 4 =a

x 3 mod 5

x 5 mod 7

x 2 mod 2

572 = 70

x= 68 + 70k

k

a1, a2,...,an

a

1 b n +1

a

b

a1, a2,...,an; a, bc:= abab:=a

b + 1

nd:= aiai:= cc:= d

an+1:=c

a1,...,an+1


27/105

t R

t > 0

n

(1 + t)n 1 + tn.

((p q) (q r)) (p r).

f : Z7 Z7

f(a) =3a + 1

Z7

f

g: Z7 Z7

g f

(g f)(a) =g(f(a))

x4 mod 5

x5 mod 7

x2 mod 2

a1, a2,...,an

a

1 b n +1

a

b


28/105

t R

t > 0

n

(1 + t)n 1 + tn.

n= 1

1 + t 1 + t.

n

n + 1

(1 + t)n+1 = (1 + t)n(1 + t) (1 + tn)(1 + t)

(1 + tn)(1 + t) = 1 + t + tn + t2

n.

1 + t + tn + t2n 1 + t(n + 1) = 1 + tn + t.

t2n 0,

t >0

((p q) (q r)) (p r).

p:= V

r:= V

q:= V

(p q) :=V

(q r) :=V

((p q) (q r)) :=V .


29/105

(p r) :=V

V F

U

A

U A

P((x, y)) :=el alumno x ha aprobado la asignatura y.

xUy A (P(x, y)).

x Uy A P(x, y).

f : Z7 Z7

f(a) =3a + 1

Z7

f

f(0) = 1f(1) = 4f(2) = 0f(3) = 3f(4) = 6f(5) = 2f(6) = 5

f

g:Z7

Z7

g f

(g f)(a) =g(f(a))

f

g

g(b) = 5b + 2

g(f(a)) =g(3a + 1) = 5(3a+ 1) + 2 =a + 5 + 2 = a


30/105

x 4 mod 5

x 5 mod 7

x 2 mod 2

572 = 70

x= 54 + 70k

k

a1, a2,...,an

a

1 b n +1

a

b

a1, a2,...,an; a, bc:= abab:=a

b + 1

nd:= aiai:= cc:= d

an+1:=c

a1,...,an+1


31/105


32/105

A3 :=

20 37 29 37 28 21 18 46 29 33 31 21 33 45 2837 38 49 63 41 42 34 62 31 41 43 36 33 65 3829 49 40 57 39 33 25 61 36 43 48 28 42 60 4037 63 57 72 60 44 41 80 46 55 61 45 54 80 5928 41 39 60 40 40 29 60 29 40 48 35 33 59 4321 42 33 44 40 26 24 52 27 30 37 30 34 51 5918 34 25 41 29 24 16 40 20 25 35 19 24 36 3246 62 61 80 60 52 40 74 40 54 65 39 52 78 5829 31 36 46 29 27 20 40 22 28 35 19 29 45 2733 41 43 55 40 32 25 54 28 32 38 26 32 55 35

31 43 48 61 48 37 35 65 35 38 44 39 35 65 4521 36 28 45 35 30 19 39 19 26 39 20 25 37 3633 33 42 54 33 34 24 52 29 32 35 25 28 53 3045 65 60 80 59 51 36 78 45 55 65 37 53 76 5928 38 40 59 43 39 32 58 27 35 45 36 30 59 40

A

A

G

G

p

q

G

g

g

G


33/105


34/105

A3 :=

20 37 29 37 28 21 18 46 29 33 31 21 33 45 2837 38 49 63 41 42 34 62 31 41 43 36 33 65 3829 49 40 57 39 33 25 61 36 43 48 28 42 60 4037 63 57 72 60 44 41 80 46 55 61 45 54 80 5928 41 39 60 40 40 29 60 29 40 48 35 33 59 4321 42 33 44 40 26 24 52 27 30 37 30 34 51 5918 34 25 41 29 24 16 40 20 25 35 19 24 36 3246 62 61 80 60 52 40 74 40 54 65 39 52 78 5829 31 36 46 29 27 20 40 22 28 35 19 29 45 2733 41 43 55 40 32 25 54 28 32 38 26 32 55 35

31 43 48 61 48 37 35 65 35 38 44 39 35 65 4521 36 28 45 35 30 19 39 19 26 39 20 25 37 3633 33 42 54 33 34 24 52 29 32 35 25 28 53 3045 65 60 80 59 51 36 78 45 55 65 37 53 76 5928 38 40 59 43 39 32 58 27 35 45 36 30 59 40

1515

(6, 9, 8, 12, 8, 7, 5, 12, 6, 7, 9, 6, 7, 12, 8).

122/2 = 61

A

A

G

A3

(1, 3)

A3


35/105

1a13

13a3

(1, 3)

A

G

p

q

p= 3/15 q= 1 p= 12/15.

1 (12/15)5.

G

g

g

N15

g

g(1) = 6, g(2) = 9, g(3) = 8, g(4) = 12, g(5) = 8, g(6) = 7, g(7) = 5, g(8) = 12,

g(9) = 6, g(10) = 7, g(11) = 9, g(12) = 6, g(13) = 7, g(14) = 12, g(15) = 8.

(6 + 9 + 8 + 12 + 8 + 7 + 5 + 12 + 6 + 7 + 9 + 6 + 7 + 12 + 8)/15.

G

K15

(15 14)/2.

|V|

G

105 61


36/105

Z6

Z6={0, 1, 2, 3, 4, 5},

f : Z6 Z6

f(x) = 2x.

f


37/105

A=

0 1 1 1 0 0 11 0 0 1 0 0 0

1 0 0 1 0 0 01 1 1 0 0 1 10 0 0 0 0 1 00 0 0 1 1 0 11 0 0 1 0 1 0


38/105

N7

R

a, b N7, (aRb (a + b)< 5) .


39/105

a :=

n :=

e:=

a (n e)a (a e)n

a

V

a e

V

a

a

F

a := V

a:= V

n:= F

e


40/105

z= 1 + ... + n= n(n 1)/2.

x 1 mod 2x 1 mod 3x 2 mod 7

2 3 7 = 42

37 + 42k k Z.

37+42 = 79

Z6

Z6={0, 1, 2, 3, 4, 5},

f : Z6 Z6

f(x) = 2x.

f

f(0) = 0

f(3) = 0

a Z6

g(a) = 1


41/105

11!

10!

10!/11! = 1/11

2 9!

10 2 9!

10 2 9!/11! = 2/11.

0.0015

1 0.9995

A=

0 1 1 1 0 0 1

1 0 0 1 0 0 01 0 0 1 0 0 01 1 1 0 0 1 10 0 0 0 0 1 00 0 0 1 1 0 11 0 0 1 0 1 0


42/105

N85

N84

N4

E :={(1, 1), (2, 2), (3, 3), (4, 4)}


43/105

E:={(1, 2), (1, 2), (3, 3), (4, 4)}

E:={(1, 2), (1, 2), (3, 4), (3, 4)}E:={(1, 2), (2, 3), (3, 4), (4, 1)}E :={(1, 2), (2, 3), (3, 1), (4, 4)}

N7

R

a, b N7, (aRb (a + b)< 5) .

N7

R:= {(1, 1), (1, 2), (2, 1), (1, 3), (3, 1), (2, 2)}.

(7, 7) / R

a+ b < 5

b+a < 5

(1, 2), (2, 1) R

(3, 1), (1, 3) R

(3, 3) /R

Ct = R {(3, 3)}


44/105

P(x)

Q(x)

x 2(mod3)x 5(mod7)

x 1(mod5)

n N


45/105

0 1 0 0 1 01 0 1 0 0 10 1 0 1 0 00 0 1 0 1 01 0 0 1 0 00 1 0 0 0 0

p = 0.1

G=(V, A)

0 1 0 0 0 0 0 01 0 1 1 0 0 0 00 1 0 1 0 0 0 0

0 1 1 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 1 0 1 00 0 0 0 0 1 0 10 0 0 0 0 0 1 0


46/105

G

(1, 5)

(V, A {(1, 5)})

G

(V, A {(1, 5)})

A= {1, 2, 3, 4, 8, 7, 54, 34, 52, 21, 11, 12, 13, 16, 112}.

A

a, b A

aRb

a b0(mod3)

R

A

A

R


47/105

P(x)

Q(x)

x U(P(x) Q(x))

xU (P(x) Q(x))

x U(P(x) Q(x))


48/105

x 2(mod3)x 5(mod7)x 1(mod5)

3 7 5 = 105

x= 26 + 105k k Z.

n N

c:= 1 + (1 + 2) + (1 + 2 + 3) + ... + (1 + 2 + ... + n)

j

n

i

j

0 1 0 0 1 01 0 1 0 0 10 1 0 1 0 00 0 1 0 1 01 0 0 1 0 00 1 0 0 0 0


49/105

p = 0.1

p= 0.1

6i=2

6

i

pi(1 p)1i.

(1, 2)

(1, 5), (5, 4), (4, 3), (3, 2)

p(1 (1 p)4).

G=

(V, A)

0 1 0 0 0 0 0 01 0 1 1 0 0 0 00 1 0 1 0 0 0 00 1 1 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 1 0 1 00 0 0 0 0 1 0 10 0 0 0 0 0 1 0

G

(1, 5)

(V, A {(1, 5)})


50/105


51/105

42n 1

n


52/105

42n 1

n

42 1 = 15

t Z

42n 1 = 15t.

42(n+1) 1

42(n+1) 1 = 42n+2 1 = 42 42n 1.

42(n+1) 1 = 42 (15t + 1) 1 = 42 15t + 16 1 = 42 15t + 15.

a1,...,anc:= 0

i= 2

n

j = 1

i 1

aj =ai

c:= j

j :=n+ 1

i:= n + 1

c:= 0

c:=

c


53/105

i 1

j

T(n)

T(n) = 3 + 3 2 + 2 + 3 3 + 2 + 3 4 + 2 + .. + 3 n + 2 + 1

c

i

O(T(n)) =O(n2).


54/105

1 + 2 + 2

2

+ ... + 2

n

= 2

n+1

1


55/105

1 + 2 + 2

2

+ ... + 2

n

= 2

n+1

1

n= 1

1 + 2 = 22 1

1 + 2 + ... + 2n + 2n+1 = (1 + 2 + .. + 2n) + 2n+1,

1 + 2 + ... + 2n

+ 2n+1

= 2n+1

1 + 2n+1

= 2n+2

1.

a1

anS := 0

i= 2

n

S := 0if ai < ai 1 S:= 1

S= 0

r:=

r:= ai1

r


56/105

G= (V, E)

0 0 1 1 0 0 0 0 0 1 0 0 0 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 0 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 1 00 0 0 0 0 0 0 0 0 0 1 0 1 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 0

G


57/105

H = (V, E)

V = V

E =

E {{10, 2}, {2, 9}}

H

H

H

H

102


58/105

G=(V, E)

0 0 1 1 0 0 0 0 0 1 0 0 0 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 0 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 1 1 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 1 00 0 0 0 0 0 0 0 0 0 1 0 1 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 0

4, 1, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2.


59/105

(Vi, Ei)

i {1, 2, 3}

V1 = {1, 4, 3, 10, 15}, V2 = {2, 5, 6, 7, 8}, V3={9, 11, 12, 13, 14}

E1 = {{1, 3}, {1, 15}, {1, 10}, {10, 3}, {4, 3}, {15, 4}, {1, 4}}

E2={{2, 5}, {6, 5}, {6, 7}, {7, 8}, {8, 6}}

E3={{11, 9}, {11, 12}, {12, 9}, {11, 14}, {12, 13}, {13, 14}}

G

{10, 2}

{2, 9}

4, 3, 10, 1, 15, 4

11, 9, 12, 13, 14, 11

H = (V

, E

)

V

= V

E

=E {{10, 2}, {2, 9}}

H

H


60/105

{6, 8}

{3, 10}

{3, 4}

{15, 4}

{9, 12}

{11, 14}

H

{6, 8}

{6, 7}

H

102

{10, 2}

{2, 5}

2 102

{2, 5}

{5, 6}

{6, 7}

{6, 8}

{7, 8}

2 102 + 102(2 102(1 102) + 104)


61/105

G= (V, E)

0 1 1 1 0 0 0 0 0 1 0 0 0 1 1

1 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 1 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 0 1 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 0

G

G


62/105

G

G

n

7n

11n 1

G = (V, E)

|V| = 30

a1

an

b

p= 1

a1

an

b

p

ai, b Z

p N {1})

b

p

A

B

A:= (p q) (r s)

B:= (p r) (q s)

A

B

G = (V, E)

M

M=

0 1 0 11 0 0 10 0 0 11 1 1 0


63/105

V

G

R

v, w V

vRw

G

v

w

R

M

k


64/105

G= (V, E)

0 1 1 1 0 0 0 0 0 1 0 0 0 1 1

1 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 1 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 0 1 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 0

G

G

G


65/105

G

G

G

{3, 10}

{3, 4}

{4, 15}

{7, 8}

{11, 14}

{11, 12}

n

7n

11n 1

71 = 7

n= 1

7n

k

7n =2k+ 1

7n+1 = 7 7n = 7 (2k+ 1) = 2 (7k+ 3) + 1.

111 1 = 10 = 2 5

n= 1

11n 1

k

11n = 5k+ 1

11n+1 1 = 11 11n 1 = 11(5k+ 1) 1 =

11 5k+ 11 1 = 55k+ 10 = 5(11k+ 2).

G = (V, E)

|V| = 30


66/105

K30

15 29

a1

an

b

p = 1

a1

an

b

p

ai, b Z

p N {1})

b

p

a1

an

b

p

j := 0

i= 1

n

ai b mod p= 0

j :=j + 1

cj :=ai

j = 0

r :=

r:= c1,...,cj

r

n

n

A

B

A:= (p q) (r s)

B:= (p r) (q s)

A

B

A

B

p, q, r, s

A

B

B

pr

q s

p

r

q

s

A

p

r


67/105

q

s

A

A

G = (V, E)

M

M=

0 1 0 11 0 0 10 0 0 11 1 1 0

V

G

R

v, w V

vRw

G

v

w

R

M

M2

M2

M2

M=

2 1 1 11 2 1 11 1 1 01 1 0 3

1 1 1 11 1 1 11 1 1 01 1 0 1

1R2

2R1

3R2

2R4


68/105

k

4

3

7

2

7 4 +

7

2

4

2

+

7

3

4

3

+

7

4

4

4

7

2

4

2

+

7

3

4

1

+

7

4

7

2

4

2

6 3


69/105

n

5n

21n 1

G =(V, E)


70/105

0 1 1 1 0 0 0 0 0 1 0 0 0 1 11 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 1 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 0 1 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 0

G

G

G

G

G = (V, E)

|V| = 40

a1

an

b

p= 1

a1

an

b

p

ai, b Z

p N {1})

b

p

G = (V, E)

M


71/105

M=

0 1 0 11 0 0 10 0 0 11 1 1 0

V

G

R

v, w V

vRw

G

v

w

R

M

A

B

A:= (t q) (u s)

B:= (t u)(q s)

A

B

k


72/105

n

5n

21n 1

51 = 5

n= 1

5n

k

5n =2k+ 1

5n+1 = 5 5n = 5 (2k+ 1) = 2 (5k+ 2) + 1.

211 1 = 20 = 4 5

n= 1

21n 1

k

21n = 5k+ 1

21n+1 1 = 21 21n 1 = 21(5k+ 1) 1 =

21 5k+ 21 1 = 21 5k+ 20 = 5(21k+ 4).

G= (V, E)


73/105

0 1 1 1 0 0 0 0 0 1 0 0 0 1 11 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 1 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 01 0 0 0 0 0 0 0 0 0 1 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 0

G

G

G

G

G


74/105

G

{3, 10}

{3, 4}

{4, 15}

{7, 8}

{11, 14}

{11, 12}

G = (V, E)

|V| = 40

K40

20 29

a1

an

b

p = 1

a1

an

b

p

ai, b Z

p N {1})

b

p

a1

an

b

p

j := 0

i= 1

n

ai b mod p= 0

j :=j + 1

cj :=ai

j = 0

r :=

r:= c1,...,cj

r

n

n

G

= (V

, E

)

M

M=

0 1 0 11 0 0 10 0 0 11 1 1 0


75/105

V

G

R

v, w V

vRw

G

v

w

R

M

M2

M2

M2

M=

2 1 1 11 2 1 11 1 1 01 1 0 3

1 1 1 11 1 1 11 1 1 01 1 0 1

1R2

2R1

3R2

2R4

A

B

A:= (t q) (u s)

B:= (t u)(q s)

A

B

A

B

t,q,u,s

A

B

B

tu

q s

t

u

q

s

A


76/105

t

u

q

s

A

A

k

72

7 4 +

7

2

4

2

+

7

3

4

3

+

7

4

4

4

7

2

4

4

+

7

3

4

3

+

7

4

4

2

+

7

5

4

1

+

7

6

7

2

4

2

6 3


77/105

n

ni=11

i(i + 1)=

n

n+ 1

G= (V, E)

0 0 1 1 0 0 0 0 0 1 0 0 0 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 1 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 1 0 0 0

1 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 1 0 0


78/105

G

G

G

G

G

n

n

a

b

b >1

n= ab

A1 := (p q)A2 := (r q)A3 :=r

B:=p

A1

A2

A3

B

G = (V, E)

M

V


79/105

M=

0 1 0 1 01 0 0 1 00 0 0 1 01 1 1 0 00 0 0 0 0

R

u

v

R

V/R

p= 1/3

0.8

n

n n


80/105


81/105

n

ni=11

i(i + 1)=

n

n+ 1

n= 1

1i=11

i(i + 1)= 1/2 = 1/(1 + 1).

n

n + 1

n+1i=11

i(i + 1)= ni=1

1

i(i + 1)+

1

(n + 1)(n+ 2).

n+1i=1 1i(i + 1)= n

n + 1+ 1

(n + 1)(n+ 2).

G= (V, E)


82/105

0 0 1 1 0 0 0 0 0 1 0 0 0 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 1 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 1 0 1 1 1 0 0 0 0 0 00 0 0 0 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 1 1 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 1 0 1 00 0 0 0 0 0 0 0 1 0 1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 1 0 0

G

(5, 1, 3, 3, 2, 4, 2, 2, 3, 2, 3, 3, 2, 2, 3).

G

G

(5, 1, 3, 3, 2, 4, 2, 2, 3, 2, 3, 3, 2, 2, 3).

G

G


83/105

(13, 15), (11, 12), (3, 4)

(3, 10), (1, 15), (6, 7).

V

V :={1, 2, 3, 4}.

E

E=

E={(1, 2)}

E={(1, 2), (1, 3)}

E={(1, 2), (4, 3)}

E={(1, 2), (1, 3), (2, 4)}

E={(1, 2), (1, 3), (1, 4)}

E={(1, 2), (1, 3), (2, 3)}

E={(1, 2), (1, 3), (2, 4), (3, 4)}

E={(1, 2), (1, 3), (2, 4), (1, 4)}

E={(1, 2), (1, 3), (2, 4), (1, 4), (3, 4)}

K4

n

n

a

b

b >1

n= ab

n

v:= 0

i = 2

n 1

n

v= 0

n mod i= 0

i:= n + 1

j :=i

j < n

j :=j j


84/105

j =n

v= 1

i:= i + 1

v= 0

s:

s:=

s

i= 2

9mod 2 = 0

i= 3

9mod 3 = 0

j = 3

j


85/105

p q

p

q

p q

p q

p q

G = (V, E)

M

V

M=

0 1 0 1 0

1 0 0 1 00 0 0 1 01 1 1 0 00 0 0 0 0

R

u

v

V ={1, 2, 3, 4, 5}

R:= {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 2), (2, 1), (1, 3),(3, 1), (1, 4), (4, 1), (2, 3), (3, 2), (2, 4), (4, 2), (3, 4), (4, 3)}

R

R

a

b

b

a

a

b

b

c

a

c

I+ M+ M2 + M3 + M4


86/105

1 1 1 1 01 1 1 1 01 1 1 1 01 1 1 1 00 0 0 0 1

V/R

V/R= {1, 5}.

p= 1/3

(2/3)3

1(2/3)3

0.8

n

p(n) = 1 23

n

.

p(n) 0.8

q(n) = ( 23

)n 0.2

q(1), q(2),...

n= 4


87/105


88/105

93

n

n n

n2

n

n

n

2n+ 2


89/105

p

q

(pq)

pq

n 2

(p1 ... pn)

p1 ... pn

p1

pn

a1

an

n

Z11

Z11

234 + 2798

34128

2765 7895943021

5x + 3 7 mod 11.


90/105

p

q

(p q)

p q

n 2

(p1 ...pn)

p1 ... pn

p1

pn

n= 2

p1 ... pn pn+1

(p1 ... pn) pn+1

(p1 ... pn pn+1)

((p1 ... pn) pn+1).

(p1 ... pn) pn+1.

(p1 ... pn) pn+1.

p1 ... pn pn+1.


91/105


92/105

234 + 2798 4 + 3 2 mod 5

34128 (1)128 1 mod 5

2765 7895943021 0 mod 5

5x 4 mod 11.

5 9 1 mod 11

x 3 mod 11.

{11k+ 3 : k Z}.


93/105

n N,

2nk=1

(2k 1) = 4n2.

x 1 mod 5

x 5 mod 11x 2 mod 3,


94/105

n N,

2nk=1

(2k 1) = 4n2.

n= 1,

2nk=0

(2k 1) =2

k=1

(2k 1) = 1 + 3 = 4 = 4n2.

2nk=1

(2k 1) = 4n2.

2(n+1)k=1

(2k 1) = 4(n + 1)2.

2(n+1)k=1

(2k 1) =2n+2k=1

(2k 1) =2nk=1

(2k 1)+[2(2n+1) 1]+[2(2n+2)1]

= 4n2 + (4n+ 1) + (4n + 3) = 4n2 + 8n+ 4 = 4(n+ 1)2.

n N,

2nk=1

(2k 1) = 4n2.


95/105

a1, a2, a3, , an.

i n, ai

7= 4i:= i + 1

i > n

s:=

s:= ai

s

n,

n

x 1 mod 5

x 5 mod 11

x 2 mod 3,

P = 3 5 11 = 165.

x0 = 1 q1 33 + 5 q2 15 + 2 q3 55,

33 q1 1 mod 5, 15 q2 1 mod 11, 55 q3 1 mod 3,

3 q1 1 mod 5, 4 q2 1 mod 11, q3 1 mod 3.

q1 = 2, q2 = 3

q3 = 1

x0= 1 2 33 + 5 3 15 + 2 1 55 =


96/105

= 66+225+110 = 66+60+110 mod 165 = 236 mod 165 = 71 mod 165.

x= 71 + 165k,

k Z.

S

S

S

S :

p(S|S) =p(S S)

p(S) .

S = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

S S = {(2, 4), (4, 2)}.

p(S|S) =p(S S)

p(S) =

2

5.

9 8 8 7 = 4032


97/105

n

nk=1

(2k)3 = 2n2(n+ 1)2.

a1,

an,

a1,

an

a1,

an


98/105

b

c

a1

an

a1

an

b

c

p

q

O1

O2

O3

P1

P2

P3

P4

P5

(O1, P1), (O2, P2), (O2, P4), (O2, P5), (O3, P3).

R

N7

R= {(1, 1), (2, 2), (4, 5), (3, 3), (4, 4), (5, 4), (5, 5), (6, 6),

(5, 6), (2, 3), (4, 6), (6, 5), (6, 4), (7, 7), (3, 2)}.


99/105

2143 + 80100

224 2513

27

312013

G = (V, E)

A

A :=

0 1 1 0 1 1 0 01 0 1 0 1 1 0 0

1 1 0 1 1 0 0 00 0 1 0 1 0 0 01 1 1 1 0 1 1 01 1 0 0 1 0 1 00 0 0 0 1 1 0 00 0 0 0 0 0 0 0

G

G

G

G


100/105

a :=

s :=

e:=

H1 := (a s),

H2:= (e s),

H3:=e.

T :=a

(H1 H2 H3) T .

a:= F, s:= F, e:= v.


101/105

F

P

fF

p P

P(p, f) :=

p

f

L(p) :=

p

fFp P (P(p, f) L(p)).

fFp P (P(p, f) L(p)).

n

nk=1

(2k)3 = 2n2(n+ 1)2.

23

= 2 1 22

n+1k=1(2k)3 = nk=1(2k)

3 + 23(n + 1)3.

n+1k=1(2k)3 = 2n2(n+ 1)2 + 23(n+ 1)3.

2(n+ 1)2

2(n+ 1)2(n2 + 22(n + 1)) = 2(n+ 1)2(n + 2)2

a1,

an,

a1,


102/105

an

a1,

an

a1,...,an

as:= 0

i= 1

n

s:= 0

a= ai

s:= 1

s

b

a1,...,anb1 := a1

j := 2

i= 2

n

pertenece(ai, b) = 0

bj :=ai

j :=j + 1

b

b

c

a1

an

a1

an

b

c

a1,...,an

b

cj:= 1

i= 1

n

ai mod c= 0

ai mod b= 0

bj :=ai

j :=j + 1

b

p

q


103/105


104/105

(5, 6), (2, 3), (4, 6), (6, 5), (6, 4), (7, 7), (3, 2)}.

R

.

(x, y)R,

(y, x) R.

(2, 3)

(3, 2)

R.

(4, 5), (5, 6) R

(4, 6)R, (4, 6), (6, 5) R

(4, 5) R,

(6, 5), (5, 4) R

(6, 4)R, (6, 4), (4, 5) R

(6, 5) R.

R

R

C1 = {1}, C2={2, 3}, C4={4, 5, 6}, C7 = {7}.

2143 + 80100 3 mod 5

224 2513 4 mod 5 3mod 5 2 mod 5

27 (22)3 2 3 mod 5

312013 343003

3 3 mod 5

G = (V, E)

A


105/105

A :=

0 1 1 0 1 1 0 01 0 1 0 1 1 0 01 1 0 1 1 0 0 00 0 1 0 1 0 0 01 1 1 1 0 1 1 01 1 0 0 1 0 1 00 0 0 0 1 1 0 00 0 0 0 0 0 0 0

G

G

G

G

v5, v4, v3, v2, v1, v3, v5, v2, v6, v5, v7, v6, v1, v5

exámenes de matemática discreta

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