Discrete Mathematics for CS
Jan 11, 2016
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Logistics
• Instructor: Meghana Nasre.
• TAs: Sreekanth, Amit, Nikhil.
• Venue: CS 26.
• Slot: A1
[ Mon: 8.00AM – 8.50AM, Th: 10.00AM – 11.50AM, Fri: 10.00AM – 10.50AM.]
• URL:
http://theory.cse.iitm.ac.in/drona/home.php?courseid=74
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Some familiar stuff..
Example 1
• 0 + 1 + 2 + 3 + ... + n
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Some familiar stuff..
Example 1
• 0 + 1 + 2 + 3 + ... + n
• What is the closed form expression for
Pn
i=0 i
?
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Some familiar stuff..
Example 1
• 0 + 1 + 2 + 3 + ... + n
• What is the closed form expression for
∀n ≥ 0,
Pn
i=0 i =
Pn
i=0 i
?
n(n+1)
2
• How do we prove it?
• Induction: a useful proof technique.
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Some familiar stuff..
Example 1
• 0 + 1 + 2 + 3 + ... + n
• What is the closed form expression for
∀n ≥ 0,
Pn
i=0 i =
Pn
i=0 i
?
n(n+1)
2
• How do we prove it?
• Induction: a useful proof technique.
• How about trying to prove
Pn
i=0 i
=
n(n−1)
2
?
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Example 2
Prove using induction:
∀n ≥ 0, n(n + 1) is odd
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Example 2
Prove using induction:
∀n ≥ 0, n(n + 1) is odd
• Assume the claim to be true for some k.
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Example 2
Prove using induction:
∀n ≥ 0, n(n + 1) is odd
• Assume the claim to be true for some k.
• Assuming this, prove it for k + 1.
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Example 2
Prove using induction:
∀n ≥ 0, n(n + 1) is odd
• Assume the claim to be true for some k.
• Assuming this, prove it for k + 1.
• Pitfall-1: Watch your base cases!
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Induction
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Example 3: Musical Chairs
• Who has NOT played musical chairs?
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Example 3: Musical Chairs
• Who has NOT played musical chairs?
• Is there some underlying principle behind musical chairs?
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Example 3: Musical Chairs
• Who has NOT played musical chairs?
• Is there some underlying principle behind musical chairs?
• More people (say n), less chairs (say k).
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Example 3: Musical Chairs
• Who has NOT played musical chairs?
• Is there some underlying principle behind musical chairs?
• More people (say n), less chairs (say k).
• How about playing it the other way?
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Example 3: Musical Chairs
• Who has NOT played musical chairs?
• Is there some underlying principle behind musical chairs?
• More people (say n), less chairs (say k).
• How about playing it the other way?
• Conclusive statements using cardinality (sizes) of these 2 sets
- set of people, set of available seats.
• Statements of the kind:
• Some person is not seated.
• All chairs are occupied.
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Example 3: Musical Chairs
• Who has NOT played musical chairs?
• Is there some underlying principle behind musical chairs?
• More people (say n), less chairs (say k).
• How about playing it the other way?
• Conclusive statements using cardinality (sizes) of these 2 sets
- set of people, set of available seats.
• Statements of the kind:
• Some person is not seated.
• All chairs are occupied.
• Why does the game of musical chairs find a winner?
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Example 4: Friends and Strangers
A
B
D
E
C
F
• 6 people. Different ways of being friends with each other.
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Example 4: Friends and Strangers
A
B
D
E
C
F
• 6 people. Different ways of being friends with each other.
• Are there always 2 friends? Are there always 2 strangers?
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Example 4: Friends and Strangers
A
B
D
E
C
F
• 6 people. Different ways of being friends with each other.
• Are there always 2 friends? Are there always 2 strangers?
• What about 3 friends OR 3 strangers?
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Example 4: Friends and Strangers
A
B
D
E
C
F
• 6 people. Different ways of being friends with each other.
• Are there always 2 friends? Are there always 2 strangers?
• What about 3 friends OR 3 strangers?
• Did we say who these people are, how they are friends?
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Example 4: Friends and Strangers
A
B
D
E
C
F
• Claim: In whichever ways these friendships are made, 6 people
will ALWAYS have either 3 friends OR 3 strangers.
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Example 4: Friends and Strangers
A
B
D
E
C
F
• Claim: In whichever ways these friendships are made, 6 people
will ALWAYS have either 3 friends OR 3 strangers.
• How do you prove this?
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Example 4: Friends and Strangers
A
B
D
E
C
F
• Claim: In whichever ways these friendships are made, 6 people
will ALWAYS have either 3 friends OR 3 strangers.
• How do you prove this?
• Try out several ways of setting up friendships. Build evidence
for truth.
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Example 4: Friends and Strangers
A
B
D
E
C
F
• Claim: In whichever ways these friendships are made, 6 people
will ALWAYS have either 3 friends OR 3 strangers.
• How do you prove this?
• Try out several ways of setting up friendships. Build evidence
for truth.
• Lets give an elegant proof
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Example 4: Friends and Strangers
A
B
D
E
C
F
• Claim: In whichever ways these friendships are made, 6 people
will ALWAYS have either 3 friends OR 3 strangers.
• How do you prove this?
• Try out several ways of setting up friendships. Build evidence
for truth.
• Lets give an elegant proof using Pigeon Hole Principle.
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More pigeons, less holes =⇒ a hole with 2 or more pigeons
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Course Overview
• Four Themes
1. Foundations: Logic and Proofs.
2. Sets, Relations, Functions.
3. Counting and Combinatorics, Discrete Probability.
4. Algebraic Structures.
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Course Overview
• Four Themes
1. Foundations: Logic and Proofs.
2. Sets, Relations, Functions.
3. Counting and Combinatorics, Discrete Probability.
4. Algebraic Structures.
• Learning outcomes include:
• Rigorous mathematical reasoning.
• Identify discrete structures and prove properties about them.
• Build necessary foundations for all courses in CS.
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Administrative Details
• A1 Slot, Venue CS 26.
• Text Book: Discrete Mathematics and its Applications
– by Kenneth Rosen, McGraw Hill, 7th Edition.
e-copy available online.
• Course Homepage: Accessible from http://theory.cse.iitm.ac.in/
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Administrative Details
• A1 Slot, Venue CS 26.
• Text Book: Discrete Mathematics and its Applications
– by Kenneth Rosen, McGraw Hill, 7th Edition.
e-copy available online.
• Course Homepage: Accessible from http://theory.cse.iitm.ac.in/
• Grading policy:
• 2 Quizzes:
20 × 2
= 40.
(Feb. 15, Mar. 21)
• End Sem:
= 45.
(Apr. 28)
• 3 Short exams:
5×3
= 15.
(Feb. 5, Mar. 11, Apr. 22)
• Ungraded practice problems at regular intervals.
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Administrative Details
• A1 Slot, Venue CS 26.
• Text Book: Discrete Mathematics and its Applications
– by Kenneth Rosen, McGraw Hill, 7th Edition.
e-copy available online.
• Course Homepage: Accessible from http://theory.cse.iitm.ac.in/
• Grading policy:
• 2 Quizzes:
20 × 2
= 40.
(Feb. 15, Mar. 21)
• End Sem:
= 45.
(Apr. 28)
• 3 Short exams:
5×3
= 15.
(Feb. 5, Mar. 11, Apr. 22)
• Ungraded practice problems at regular intervals.
• Friendly TAs: Sreekanth, Amit, Nikhil.
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Questions?
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