Induction
Requirements
1. Formally state the property that will be proved inductively.
2. Prove the property holds in the base case.
3. Formally state the inductive hypothesis.
4. Assume the inductive hypothesis, and prove the inductive step.
5. Conclude
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 1
Due: Tuesday, 4 Feb 2013
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 3
Due: Wednesday, 20 Feb 2013
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 2
Due: Tuesday, 11 Feb 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 5
Due: Tuesday, 11 Mar 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 4
Due: Tuesday, 4 Mar 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 6
Due: Tuesday, 18 Mar 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 7
Due: Wednesday, 2 Apr 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID i
Set Identities
Let all sets referred to below be subsets of a universal set U .
1. Commutative Laws: For all sets A and B,
(a) A B = B A and (b) A B = B A
2. Associative Laws: For all sets A, B, and C,
(a) (A B) C = A (B C) and (b) (A B) C = A (B C)
3. Di
Division into Cases
Requirements
1. Show that the proposition always falls into one of a few cases.
2. List the cases.
3. Under each case, give a proof that the proposition holds for that case.
4. Conclude that the overall proposition holds.
Example
Prove
Memory Circuits
CS22
March 16, 2013
Combinatorial circuits consisting of AND, OR, and NOT gates are very useful
in modeling conditions or boolean expressions. For example, the above circuits
output is 1 if and only if x2 = 1 and x1 = 0.
However, we can al
Bijection
Requirements
1. Formally dene the two sets claimed to have equal cardinality.
2. Formally dene a function from one set to the other.
3. Prove that the function is bijective by proving that it is both injective and surjective.
4. Conclude that si
Logical Equivalences
Given any statement variables p, q, and r, a tautology t and a contradiction c, the
following logical equivalences hold.
1. Commutative Laws:
p q q p and p q q p
2. Associative Laws:
(p q) r p (q r) and (p q) r p (q r)
3. Distributive
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 10
Due: Tuesday, 29 April 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 9
Due: Tuesday, 22 Apr 2013
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
CSCI 0220
Discrete Structures and Probability
C. Klivans
Homework 8
Due: Tuesday, 15 Apr 2014
All homeworks are due at 11:00PM in the CS22 bin on the CIT second oor,
next to the Fishbowl.
Write your full name and login (CS login if you have one, AuthID if
Set Equivalence
The following are two sample proofs of the equivalence
(A B) (A B) = A (B (A B).
One uses the element method. The other uses set algebra.
NOTE: Drawing a Venn diagram of each does not constitute a proof and will not be graded
as such.
Elem
More Induction Proofs
Example 1
Prove that f (n) = 6n2 + 2n + 15 is odd for all n Z+ .
Proof by induction:
Dene P (n) as the property that f (n) is odd.
Base case: We prove P (1).
2(1) + 15 + 6(1)2 = 2 + 15 + 6 = 23. Since 23 is odd, P (1) is true.
Induct