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IIT Bombay :: Autumn 2020 :: CS 207 :: Discrete Structures :: Manoj Prabhakaran
Discrete Structures A flavour Bridges of Knigsberg Cross each - - PDF document
Discrete Structures A flavour Bridges of Knigsberg Cross each - - PDF document
IIT Bombay :: Autumn 2020 :: CS 207 :: Discrete Structures :: Manoj Prabhakaran (Introduction to Mind Bending) Discrete Structures A flavour Bridges of Knigsberg Cross each bridge exactly once ?! Is it impossible? How do we know for sure?
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Bridges of Königsberg
Cross each bridge exactly once Is it impossible? How do we know for sure?
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Discrete Stuff
Graphs (maps, friendships, www…) Patterns, Symmetry Numbers Logic, reasoning (Discrete) Algorithms Digital computers...
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Pigeonholes & Parties
Suppose you go to a party and there is a game: How many of your “friends” are at the party? (Everyone who goes to the party has at least one person there that he/she counts as a friend.) There will be at least two who have the same number of friends at the party! But Why?
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Pigeonholes & Parties
Suppose you go to a party and there is a game: How many of your “friends” are at the party? (Everyone who goes to the party has at least one person there that he/she counts as a friend.) There will be at least two who have the same number of friends at the party! If there are 4 people in the party, for each person, the number of friends at the party is 1, 2 or 3. There are 4 of you, and everyone needs to pick a
- number. There are only 3 numbers to pick from...
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The Pigeonhole Principle
If there are more pigeons than pigeonholes, then at least one pigeonhole will have more than one pigeon in it
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Pigeonholes & Parties
Point to ponder Suppose friendships are always reciprocated. Then can you show that the claim holds even if not everyone has a friend at the party?
So again, suppose you go to a party and there is a game: How many of your “friends” are at the party? (Everyone who goes to the party has at least one person there that he/she counts as a friend.) There will be at least two who have the same number of friends at the party!
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The Skippy Clock
Has 13 hours on its dial! Needle moves two hours at a time Which all numbers will the needle reach? Reaches all of them!
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Points to ponder What if the clock had 12 hours? What if the needle moved 5 hours at a time?
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Topics to be covered
Basic tools for expressing ideas Logic, Proofs, Sets, Relations, Functions Numbers and patterns therein Graphs Recursion Trees Counting Induction Bounding big-O