[SOLVED] CS Induction and recursion

Induction and recursion

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Section 5.1

: A B R AB R A B
A = {0,1,2} B = {a,b}
{(0, a), (0, b), (1,a) , (2, b)} A B
A B :

AB

: A R A A A A
A = {a,b,c} R = {(a,a),(a,b), (a,c)} A
A = {1, 2, 3, 4}
R= {(a,b) | a b} (1,1), (1, 2), (1,3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)

()
: A?
:A A A, A A
A n A A n2 , n 2n , A A
n

()
: :
R1 = {(a,b) | a b},R4 = {(a,b) | a = b},
R2 = {(a,b) | a > b},R5 = {(a,b) | a = b + 1},
R3 = {(a,b) | a = b a = b}, R6 = {(a,b) | a + b 3}.

(1,1), (1, 2), (2, 1), (1, 1), (2, 2)?
:, (1,1) R1, R3, R4 , R6, (1,2) R1 R6 (2,1) R2, R5, R6 (1, 1) R2, R3, R6 (2,2) R1, R3, R4

: R iff x A(x, x) R
x[x A (x, x) R]
: :
R1 = {(a,b) | a b},R4 = {(a,b) | a = b},
R2 = {(a,b) | a > b},R5 = {(a,b) | a = b + 1},
R3 = {(a,b) | a = b a = b}, R6 = {(a,b) | a + b 3}.

A = !

: R iff x A(x, x) R
x[x A (x, x)R]
: :
R1 = {(a,b) | a b},R4 = {(a,b) | a = b},
R2 = {(a,b) | a > b},R5 = {(a,b) | a = b + 1},
R3 = {(a,b) | a = b a = b}, R6 = {(a,b) | a + b 3}.

: R x,y A(x,y) R (y,x) R xy [(x,y) R (y,x) R]
: :
R1 = {(a,b) | a b},R4 = {(a,b) | a = b},
R2 = {(a,b) | a > b},R5 = {(a,b) | a = b + 1},
R3 = {(a,b) | a = b a = b}, R6 = {(a,b) | a + b 3}.

: Rx,y A (x,y) R (y,x) R x = y xy [(x,y) R (y,x) R x = y]
::
R1 = {(a,b) | a b},R4 = {(a,b) | a = b},
R2 = {(a,b) | a > b},R5 = {(a,b) | a = b + 1},
R3 = {(a,b) | a = b a = b}, R6 = {(a,b) | a + b 3}.

: A Rx,y,z A(x,y) R (y,z) R, (x,z) R
xy z[(x,y) R (y,z) R (x,z) R ]
: :
R1 = {(a,b) | a b},R4 = {(a,b) | a = b},
R2 = {(a,b) | a > b},R5 = {(a,b) | a = b + 1},
R3 = {(a,b) | a = b a = b}, R6 = {(a,b) | a + b 3}.

(a) {(2,2), (2,3), (2,4), (3,2), (3,3), (3,4)}
(b) {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4)}
(c) {(2,4), (4,2)}
(d) {(1,2), (2,3), (3,4)}
(e) {(1,1), (2,2), (3,3), (4,4)}
(f) {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4)}

R1 R2, R1 R2, R1 R2, R1 R2, R2 R1
: A = {1,2,3} B = {1,2,3,4} R1 = {(1,1),(2,2),(3,3)} R2 = {(1,1),(1,2),(1,3),(1,4)} :

R1 R2 ={(1,1),(1,2),(1,3),(1,4),(2,2),(3,3)}
R1 R2 ={(1,1)}
R1 R2 ={(2,2),(3,3)}
R2 R1 ={(1,2),(1,3),(1,4)}

R1 A B
R2 B C
R1R2, A C
(x,y) R1 (y,z)R2, (x,z) R2 R1

R2 R1= {(b,x),(b,z)}

R={(1,1),(1,4),(2,3),(3,1),(3,4)}
S={(1,0),(2,0), (3,1),(3,2),(4,1)}

A={a, b, c, d}R1R2A
R1={(a,a), (a,b), (b,d)},
R2={(a,d), (b,c), (b,d), (c,d)}
R1oR2R2oR1

: R A R Rn :
: R1 = R
:Rn+1 = Rn R
:
1: AR Rn R n = 1,2,3 .

1(a,b) Rn+1 Rn+1 = Rn Rx A(a,x) R(x,b) Rn Rn R(x,b) R
2R (a,x) R(x,b) R(a,b) R

R={(a,b),(b,c),(c,d),(d,b)}

Section 5.3

0-1
R A = {a1, a2, , am} B = {b1, b2, , bn}
R MR = [mij]

ai bj R0-1(i, j)1 0

1: A = {1,2,3} B = {1,2}R A B a A,b B, a > bR(a, b) R ?
: R = {(2,1), (3,1),(3,2)},

()

(cont.)
2: A = {a1,a2, a3} B = {b1,b2, b3,b4, b5}R

: R (ai, bj) mij = 1, :
R = {(a1, b2), (a2, b1),(a2, b3), (a2, b4),(a3, b1), {(a3, b3), (a3, b5)}.

R , MR 1

R , mij = 1mji = 1
R , mij = 1 i j mji = 0

3: R

R , , ?
: 1, R MR , R

: V E V a (a,b) b
(a,a)
7:a, b, cd,
(a, b), (a, d), (b, b), (b, d), (c, a), (c, b)
(d, b)

8:?

:
(1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 3),(4, 1)(4, 3)

:

: (x,y), (y,x)
: (x,y) x y, (y,x)
: (x,y) (y,z), (x,z)

?
? ()
? (Triviality)
? , (Triviality)

1

Triviality()

?
a b, b a
? db b d
? a b b d, a d

2

?
c a
,
? a b

3

?
d a
,
? (trivially),

4

A={1,2,3}

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