extensible networking platform 1 1 - cse 240 – logic and discrete mathematics review: mathematical...
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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 Tiling
A 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 1
0 if 0
)(
nnfnf
n
n
nf