Discrete Structures

Lecture Notes, 8 Nov 2010

 Last Update: 8 November 2010 Note: or material is highlighted

§4.1: Mathematical Induction

1. Function Composition:

1. Reminder:
Def:

Let A,B,C be sets.
Let g : A → B.
Let f : B → C.
Then the composition of f with g
—denoted "(f o g) : A → C" & read "f of g"—
=def {(a, c) | (∃b ∈ B)[(g(a) = b) ∧ (f(b) = c)]

1. i.e., (f o g)(a) = f(g(a))

2. E.g.:

Let f(x) = x+1.
Let g(y) = 3y.

Then: (f o g)(z) = f(g(z)) = f(3z) = 3z+1

but: (g o f)(z) = g(f(z)) = g(z+1) = 3(z+1) = 3z+3.

3. Thm: Function composition is not commutative

• i.e., (f o g) ≠ (g o f)

2. Thm:

Let A,B be sets.
Let f : A → B be a 1– correspondence

(i.e., total, 1–1 & onto)

Then (f o f–1) : B → B = ιB
and (f–1 o f) : A → A = ιA
and (f–1)–1 = f

2. Sequences:

1. Def:

Let (S = N) ∨ (S = W (= Z+)).
Let T be any non-∅ set.
Then a is a sequence =def a : S → T.

• i.e., a sequence is a function from {(0,) 1, 2, …} to any set T

2. Notation:

1. an for: a(n)
2. "an" is a term of the sequence
3. {an} for: the sequence a : N → T (or the sequence a : W → T)

3. A sequence is a function.
A function is a relation.
A relation is a set.
∴ A sequence is a set: {(0, a0), (1, a1), …}, where each ai ∈ T

4. A sequence can be thought of as an ∞-tuple

5. Examples:

1. Example 1:

0,  1,   4,   9,  16,  25, …
a0, a1, a2, a3, a4,  a5, …

i.e., (∀nN)[an = n²]

0,  1,   1,   2,   3,   5,   8,   13, …
a0, a1, a2, a3, a4,  a5, a6,   a7, …

i.e.:

a0 = 0
a1 = 1
(∀n > 1)[an = an–1 + an–2]

6. General problem:

1. Given a sequence, to determine its formula
2. i.e., given I/P & O/P of an algorithm, to determine (the?) an(!) algorithm for it.
3. Notes:

1. language learning is a real-life e.g.
2. in general, ∃ >1 algorithm!
3. given only a finite, initial sequence of O/P,
∃ no way to find the intended algorithm

• i.e., ∃ no way to predict the future!

4. See "Computational Learning Theory"

7. Summations:

1. A series or summation =def the sum of (some or all) terms in a sequence.

2. i.e.) a0 + a1 + a2 + …

3. Notation:

i = n
Σai for: a0 + a1 + … + an
i = 0

"Sequences and Summations: Further Information"

3. Mathematical Induction:

1. Watch the falling-domino movies at "Recursion & Induction"

2. Let Falls(x) mean: x falls down.
Then:

Falls(domino1);      "base case"
Falls(domino1)Falls(domino2)
∴ Falls(domino2),by Modus Ponens!
Falls(domino2)Falls(domino3)
∴ Falls(domino3),by MP
∴ Falls(dominok) "inductive hypothesis"
Falls(dominok)Falls(dominok+1); "inductive case"
∴ Falls(dominok+1)
Falls(dominolast–1)Falls(dominolast)
∴ Falls(dominolast)

n[Falls(dominon)]   "general principle"