peer to peer network properties
TRANSCRIPT
Extensible Networking Platform 1 1 - CSE 240 – Logic and Discrete Mathematics
Review: Mathematical Induction
Use induction to prove that the sum of the first n odd integers is n2.
Base case (n=1): the sum of the first 1 odd integer is 12. Yes, 1 = 12.
Assume P(k): the sum of the first k odd ints is k2. 1 + 3 + … + (2k - 1) = k2
Prove that 1 + 3 + … + (2k - 1) + (2k + 1) = (k+1)2
1 + 3 + … + (2k-1) + (2k+1) =
k2 + (2k + 1)= (k+1)2
Extensible Networking Platform 2 2 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
Deficient TilingA 2n x 2n sized grid is deficient if all but one cell is tiled.
2n
2n
Extensible Networking Platform 3 3 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
• We want to show that all 2n x 2n sized deficient grids can be tiled with tiles, called triominoes, shaped like:
Extensible Networking Platform 4 4 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
• Is it true for all 21 x 21 grids?
Extensible Networking Platform 5 5 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
Inductive Hypothesis:We can tile any 2k x 2k
deficient board using our fancy designer tiles.
Use this to prove:We can tile any 2k+1 x 2k+1
deficient board using our fancy designer tiles.
Extensible Networking Platform 6 6 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
2k
2k 2k
2k
2k+1
OK!! (by IH)
???
Extensible Networking Platform 7 7 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
2k
2k 2k
2k
2k+1
OK!! (by IH)
OK!! (by IH)
OK!! (by IH)
OK!! (by IH)
Extensible Networking Platform 8 8 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - a cool example
Extensible Networking Platform 9 9 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - why does it work?
Definition:A set S is “well-ordered” if every non-
empty subset of S has a least element.
Given (we take as an axiom): the set of natural numbers (N) is well-ordered.
Is the set of integers (Z) well ordered?No.
{ x Z : x < 0 } has no least
element.
Extensible Networking Platform 10 10 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - why does it work?
Is the set of non-negative reals (R) well ordered?
Extensible Networking Platform 11 11 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - why does it work?
Proof of Mathematical Induction:
We prove that (P(0) (k P(k) P(k+1))) (n P(n))
Assume1. P(0)2. k P(k) P(k+1)3. n P(n) n P(n)
Extensible Networking Platform 12 12 - CSE 240 – Logic and Discrete Mathematics
Mathematical Induction - why does it work?
Assume1. P(0)2. n P(n) P(n+1)3. n P(n) n P(n)
Let S = { n : P(n) }
What do we know? -P(k) is false because it’s in S. -k 0 because P(0) is true. -P(k-1) is true because P(k) is the least
element in S.
Extensible Networking Platform 13 13 - CSE 240 – Logic and Discrete Mathematics
Strong Mathematical Induction
If P(0) and n0 (P(0) P(1) … P(n)) P(n+1)
Thenn0 P(n) In our proofs, to show P(k+1), our
inductive hypothesis assumes that ALL of P(0), P(1), … P(k)
are true, so we can use ANY of them to make the inference.
Extensible Networking Platform 14 14 - CSE 240 – Logic and Discrete Mathematics
Game with Matches• Two players take turns removing any
number of matches from one of two piles of matches. The player who removes the last match wins
• Show that if two piles contain the same number of matches initially, then the second player is guaranteed a win
Extensible Networking Platform 15 15 - CSE 240 – Logic and Discrete Mathematics
Strategy for Second Player• Let P(n) denote the statement “the second
player wins when they are initially n matches in each pile”
• Basis step: P(1) is true, because only 1 match in each pile, first player must remove one match from one pile. Second player removes other match and wins
• Inductive step: suppose P(j) is True for all j 1<=j <= k.
• Prove that P(k+1) is true, that is the second player wins when each piles contains k+1 matches
Extensible Networking Platform 16 16 - CSE 240 – Logic and Discrete Mathematics
Strategy for Second Player• Suppose that the first player removes
r matches from one pile, leaving k+1 –r matches there
• By removing the same number of matches from the other pile the second player creates the situation of two piles with k+1-r matches in each. Apply the inductive hypothesis and the second player wins each time.
Extensible Networking Platform 17 17 - CSE 240 – Logic and Discrete Mathematics
Postage Stamp Example• Prove that every amount of postage
of 12 cents or more can be formed using just 4-cent and 5-cent stamps
• P(n) : Postage of n cents can be formed using 4-cent and 5-cent stamps
• All n >= 12, P(n) is true
Extensible Networking Platform 18 18 - CSE 240 – Logic and Discrete Mathematics
Postage Stamp Proof• Base Case: n = 12, n = 13, n = 14, n = 15
– We can form postage of 12 cents using 3, 4-cent stamps– We can form postage of 13 cents using 2, 4- cent stamps
and 1 5-cent stamp– We can form postage of 14 cents using 1, 4-cent stamp
and 2 5-cent stamps– We can form postage of 15 cents using 3, 5-cent stamps
• Induction Step– Let n >= 15– Assume P(k) is true for 12 <= k <= n, that is postage of
k cents can be formed with 4-cent and 5-cent stamps (Inductive Hypothesis)
– Prove P(n+1)– To form postage of n +1 cents, use the stamps that form
postage of n-3 cents (from I.H) with a 4-cent stamp
Extensible Networking Platform 19 19 - CSE 240 – Logic and Discrete Mathematics
Recursive Definitions
We completely understand the function f(n) = n!, right?
As a reminder, here’s the definition:n! = 1 · 2 · 3 · … · (n-1) · n, n 1
But equivalently, we could define it like this:
0 n if 1
1n if )!1(! nnn
Extensible Networking Platform 20 20 - CSE 240 – Logic and Discrete Mathematics
Recursive Definitions
Another VERY common example:
Fibonacci Numbers
1 if )2()1(1 if 10 if 0
)(nnfnfnn
nf