Algoritmos e inducción

110 Pages·2012·1.04 MB·English

by  Luis Sierra

Algoritmos e inducci´on
Luis Sierra
InstitutodeComputaci´on
FacultaddeIngenier´ıa
UniversidaddelaRepu´blica
Febrero del 2012
Pr´acticas compartidas para la ensen˜anza de inform´atica:
el aula y el trabajo
Universidad Nacional de Quilmes, Argentina
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 1/40
Outline
1 Inducci´on como justificaci´on
2 Inducci´on como descubrimiento
3 Sumar
4 Tipos de datos
5 Programaci´on
6 Visualizaci´on
7 Conclusiones
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 2/40
Inducci´oncomojustificaci´on
n
n+1
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 3/40
Inducci´oncomojustificaci´on
Gauss
n
(cid:88) n(n+1)
i =
2
i=0
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 4/40
We verify that the formula holds true for the smallest possible value
If we know that the formula holds true for the numbers < N, then it
also holds true for the number N.
From that we conclude that the formula holds true for all the
numbers.
Inducci´oncomojustificaci´on
Induccio´n como justificaci´on
n
(cid:88) n(n+1)
i =
2
i=0
Proof by Mathematical Induction follows this pattern: We want to verify
that a formula, algebraic expression, holds true for all the values of the
parameter, a whole number.
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 5/40
If we know that the formula holds true for the numbers < N, then it
also holds true for the number N.
From that we conclude that the formula holds true for all the
numbers.
Inducci´oncomojustificaci´on
Induccio´n como justificaci´on
n
(cid:88) n(n+1)
i =
2
i=0
Proof by Mathematical Induction follows this pattern: We want to verify
that a formula, algebraic expression, holds true for all the values of the
parameter, a whole number.
We verify that the formula holds true for the smallest possible value
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 5/40
From that we conclude that the formula holds true for all the
numbers.
Inducci´oncomojustificaci´on
Induccio´n como justificaci´on
n
(cid:88) n(n+1)
i =
2
i=0
Proof by Mathematical Induction follows this pattern: We want to verify
that a formula, algebraic expression, holds true for all the values of the
parameter, a whole number.
We verify that the formula holds true for the smallest possible value
If we know that the formula holds true for the numbers < N, then it
also holds true for the number N.
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 5/40
Inducci´oncomojustificaci´on
Induccio´n como justificaci´on
n
(cid:88) n(n+1)
i =
2
i=0
Proof by Mathematical Induction follows this pattern: We want to verify
that a formula, algebraic expression, holds true for all the values of the
parameter, a whole number.
We verify that the formula holds true for the smallest possible value
If we know that the formula holds true for the numbers < N, then it
also holds true for the number N.
From that we conclude that the formula holds true for all the
numbers.
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 5/40
(cid:80)n+1i = (n+1)(n+2)
i=0 2
⇐=
(cid:80)0 = 0(0+1)
i=0 2 n+1+(cid:80)n i = (n+1)(n+2)
⇐= i=0 2
⇐= (*)
0 = 0.
n(n+1) (n+1)(n+2)
n+1+ =
2 2
⇐=
2(n+1)+n(n+1) = (n+1)(n+2).
Inducci´oncomojustificaci´on
Induccio´n como justificaci´on
(∀n∈N::P.n)
n
(cid:88) n(n+1)
P.n := i =
2
i=0
Paso: (∀n ∈ N : P.n : P.(n+1))
Base: P.0
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 6/40
(cid:80)n+1i = (n+1)(n+2)
i=0 2
⇐=
n+1+(cid:80)n i = (n+1)(n+2)
i=0 2
⇐= (*)
n(n+1) (n+1)(n+2)
n+1+ =
2 2
⇐=
2(n+1)+n(n+1) = (n+1)(n+2).
Inducci´oncomojustificaci´on
Induccio´n como justificaci´on
(∀n∈N::P.n)
n
(cid:88) n(n+1)
P.n := i =
2
i=0
Paso: (∀n ∈ N : P.n : P.(n+1))
Base: P.0
(cid:80)0 = 0(0+1)
i=0 2
⇐=
0 = 0.
LuisSierra (InCo-Uruguay) Algoritmoseinduccio´n Quilmes2012 6/40