Week 2 notes

The primary goal of this course is to become familiar with some of the basic mathematical ideas used in programming.

Reading:

        Levin, 0.1 and 0.2

Negation rules:  (¬A can also be written ~A)

  A => B        same as    ¬A or B

¬(A and B)    same as   ¬A or ¬B   (if it is not true that A is true and B is true, then one of them must be false)    (Morgan's Law)

¬(A or B)      same as   ¬A  and ¬B  (if it is not true that either A or B is true, then both of them must be false)    (Morgan's Law)

Quantifiers: ∀ (for all) and ∃ (there exists)

Negation rules for quantifiers:

  ¬∃x P(x)     same as  ∀x ¬P(x)    (if there does not exist an x making P(x) true, then P(x) is false for every x)

  ¬∀x P(x)     same as  ∃x ¬P(x)    (if P(X) is not true for every x, then for some x it is false)

Here are some simple two-person two-move games. In each case, player A chooses a whole number a, and then, seeing A's move, B chooses a whole number b. Based on the choices a and b, we determine which player wins.

1. The first player to choose a 3 wins (that is, B wins if a!=3 and b=3)
2. B wins if a+b<100
3. Whichever player chooses the largest number wins

In each game above, who has the winning strategy?

Does every 2-move game have a winning strategy?

Player A has a winning strategy if ∃a ∀b (player A wins). If this is false, that mean

               ¬ ∃a ∀b (player A wins)

same as     ∀a ¬∀b (player A wins)

same as    ∀a  ∃b ¬(player A wins)

same as    ∀a  ∃b (player B wins)

Negation of quantifiers

From Levin p 15. Are any of the following equivalent? Do any imply any other?

1. You can fool some people all of the time.
2. You can fool everyone some of the time.
3. You can always fool some people.
4. Sometimes you can fool everyone.

The first is (∃ person p)(∀ time t) CanFool(p,t)
The second is either ∀p∃t(CanFool(p,t)) or else ∃t∀p CanFool(p,t). What is the difference? Let's go with the first.
The third is ∃p∀t CanFool(p,t), same as the first.
The fourth is ∃t∀p CanFool(p,t)

A predicate is a formula with a free variable. We can bind that variable by sticking a quantifier for it on the front.

A few more examples of quantifier negation:

Not exists

¬∃a ∀b Win1(a,b)

lim f(x) = L as x->a: ∀e ∃d ∀x |x-a| < d => |f(x)-L| < e.

Suppose you are driving down the highway and see a sign:

    No passing zones not striped

A hyphen is missing: no-passing zones are not striped, because they just repaved. But what if we take it to mean:

    There are not any passing zones that are not striped

What does this mean?

sqrt(2)

    start with: n is even if and only if n*n is even.

    n is even if ∃k (n=2*k)

Proving ∀n∈N ∃k∈N(n=2k or n=2k+1). Our first Proof by Mathematical Induction.

Two-move games. The moves are a1, b1, a2, b2

A has a winning strategy if ∃a1 ∀b1 ∃a2 ∀b2 (A wins).

A does not have a winning strategy if ~ ∃a1 ∀b1 ∃a2 ∀b2 (A wins)

which is    ∀a1 ∃b1 ∀a2 ∃b2 (~ A wins), which is ∀a1 ∃b1 ∀a2 ∃b2 (B wins), which means there is a winning strategy for B.

game 1: B wins if a1<b1<a2<b2. Who has the winning strategy?

game2: B wins if a1/a2 = b1/b2, or a2==b2==0. Who has the winning strategy?

game 3: B wins if exactly one of a1,a2 is between b1 and b2.

game 4: B wins if the line from (a1,b1) to (a2,b2) has positive slope (or a1=a2)

game 5: B wins if the line from (a1,a2) to (b1,b2) has positive slope

game 6: A and B pick points on the plane. B wins if b2 is equidistant from a1, b1, a2.

Sets: Levin 0.3

• membership
• Listing elements versus giving a rule
• set equality; order does not matter
• subsets
• empty set, ∅
• union and intersection
• Venn diagrams

End here Monday 1/24

Functions: Levin 0.4

• defined by table or by rule
• domain
• recursive functions f:N->N 
• bijections
• injections (1-to-1)
• surjections
• set-theoretic definition of f:A->B as a subset of AxB

Counting: Levin 1.1, on page 57

• Additive principle
• Multiplicative principle
• Set-theoretic approach
• Pies on page 64

Counting subsets of {1, 2, ..., N}, Levin 1.2 on page 70

Homework 1, due Friday Jan 28, on Sakai