1. and Structural Induction II
Mathematics for Computer Science MIT 6.042J/18.062J
Recursive Definitions
and
Structural Induction II
Copyright © Albert Meyer, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

2. … Single node is a Rooted Tree. If t1, t2, …, tn are RTs, so is: t1 t2
Rooted Trees
Single node is a Rooted Tree.
If t1, t2, …, tn are RTs, so is: … t1 t2 tn

3. Rooted Trees
In scheme:
(((())()())((())())((()))(()(()())))

4. Structural vs Ordinary Induction
Structural Induction on Rooted Trees
reduces to Strong Induction on
SIZE of subtrees.
But what is the "size" of the COSINE function?
Data objects may be infinite:
The Functions

5. Structural vs Ordinary Induction
For 18.01F, replace structural induction
by induction on number of steps
to construct an object .
Function no. of steps x, sin(x), cos(x), ex : 1
log(x) [inverse,ex] 2
cos(log(x)) [cos,log,compose] 4
x ·cos(log(x)) [id,times,] 6

6. Structural vs Ordinary Induction
derivation tree for x ·cos(log(x)) *
x cos(log x) x 
cos(log x)
cos
inv
log ex

7. Structural vs Ordinary Induction
But Structural Induction also applies to Infinite Trees

8. … … … … … Infinitely wide Game Tree for Problem 1 (0,0) (0,1) (1,0)
(1,1)
(0,2)
(2,0)
(1,2)
(2,1) … …
(0,0)
(0,0) … …
(0,0)
(0,1)
(0,2)
(0,1)
(0,2)
Infinitely wide

9. In-Class Problem
Problem 1

10. … … Single node is an FP tree If t1, t2, …, tn are FPs, so is: t1 t2
Finite-Path Trees
Single node is an FP tree
If t1, t2, …, tn are FPs, so is: … … t1 t2 tn
May have infinitely many subtrees

11. Finite-Path Trees with
Rooted Trees are
Finite-Path Trees with
finitely many nodes

12. Finite-Path Trees … …

13. Finite-Path Trees … …

14. Finite-Path Trees: Unboundedly Deep

15. Finite-Path Trees: Wide & Deep

16. But every downward path in an FP eventually stops
Finite-Path Trees
But every downward path in an FP eventually stops

17. Proof by structural induction on def. of FP
Finite-Path Trees
Proof by structural induction on def. of FP
Base Case (t is one node):
Has only a 0 length path

18. Proof by structural induction on def. of FP
Finite-Path Trees
Proof by structural induction on def. of FP
Induction Step (t has subtrees):
Assume in subtrees all down paths are finite.
Now path from root of t goes to root of subtree. Rest of the path in subtree must be finite by induction QED

19. In-Class Problem
Problem 2

20. Mathematics for Computer Science MIT 6.042J/18.062J
Sums, Products & Asymptotics
Copyright © Albert Meyer, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

21. C. F. Gauss
Picture source:

22. Sum for children … …
10,445 +
10, … ,372

23. Sum for children
Nine-year old Gauss (so the story goes) saw that each number was 103 greater than the previous one.

24. Sum for children
( ) + ( (103)) +
( (103)) + … + ( (103))
= 8900(25) + 103( … + 24)

25. Arithmetic Series
A ::= … + (n-1) + n

26. A ::= 1 + 2 + … + (n-1) + n A = 1 + 2 + … + (n-1) + n
Arithmetic Series
A ::= … + (n-1) + n
A = … + (n-1) + n

27. A ::= 1 + 2 + … + (n-1) + n A = n + (n-1) + … + 2 + 1
Arithmetic Series
A ::= … + (n-1) + n
A =
n + (n-1) + …

28. 2A =(n+1)+(n+1) + … + (n+1) + (n+1)
Arithmetic Series
A ::= … + (n-1) + n
A =
n + (n-1) + …
2A =(n+1)+(n+1) … + (n+1) + (n+1)

29. 2A =(n+1)+(n+1) + … + (n+1) + (n+1)
Arithmetic Series
A ::= … + (n-1) + n
A =
n + (n-1) + …
2A =(n+1)+(n+1) … + (n+1) + (n+1)
= n(n+1)

30. Arithmetic Series So

31. Geometric Series

32. Geometric Series

33. Geometric Series

34. Geometric Series
xn+1

35. Geometric Series
n+1

36. The future value of $$. I will promise to pay you $100
Annuities
The future value of $$.
I will promise to pay you $100
in exactly one year,
if you will pay me $X now.

37. X = $100/1.03 ≈ $97.09 My bank will pay me 3% interest.
Annuities
My bank will pay me 3% interest.
I can’t lose if you pay me:
X = $100/1.03 ≈ $97.09

38. 97.09¢ today is worth $1.00 in a year
Annuities
97.09¢ today is worth $1.00 in a year
$1.00 in a year is worth $1/1.03 today
$n in a year is worth $nr today,
where r = 1/1.03

39. $n in two years is worth $nr2 today $n in k years is worth $nrk today
Annuities
$n in two years is worth $nr2 today
$n in k years is worth $nrk today

40. I will pay you $100/year for 10 years If you will pay me $Y now.
Annuities
I will pay you $100/year for 10 years
If you will pay me $Y now.
I can’t lose if you pay me
100r + 100r r3 + … + 100r10
=100r(1+ r + … + r9)
= 100r(1-r10)/(1-r) = $853.02

41. In-Class Problem
Problems 3 & 4