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Data Structures and Algorithms
Discrete Math Review
Chris Brooks
Department of Computer Science
University of San Francisco
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2-0:
Discrete Math Review
Sets
Logarithms
Summations
Recursion
Proof Techniques
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2-1:
Sets
A set is a collection of distinguishable objects (usually
called members or elements)
{1,2,3,4}
{red, green, blue}
{1,2,3,4,5,6,...}
is in set
indicates that object
denotes the cardinality (size) of set
is the set with no elements (the “empty set”)
.
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: All the objects in A or B
All the objects in A and B
All the objects that are in A, but not
: A is a subset of B (or B is a superset of A)
Set Difference:
in B.
Intersection:
Set Operations
Union:
2-2:
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2-3:
Logarithms
(written as
In many places, logs use base 10 or base
is equivalent to:
Think of it as ’the power you’d need to raise b to to
yield y.’
Logarithms (or ’logs’) are most easily thought of as the
inverse of exponentiation.
)
In computer science, we typically use base 2
This is a result of using binary arithmetic.
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2-4:
Logarithm examples
How many bits does it take to encode 1,000,000 different
values?
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Logarithm examples
2-5:
How many bits does it take to encode 1,000,000 different
values?
Each bit can take on one of two values (1 or 0)
Therefore, bits can represent
values.
So, encoding 1,000,000 values will require
= 20
bits
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2-6:
Logarithm examples
If we have an alphabetically sorted list of 128 names, how
many records do we need to look at to find a given
individual?
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Logarithm examples
2-7:
If we have an alphabetically sorted list of 128 names, how
many records do we need to look at to find a given
individual?
Since the list is sorted, we can use binary search.
Look at the middle element: if it’s after than the name
we’re looking for, search the first half of the list. If it’s
before the name we’re looking for, look at the second
half of the list.
Each check cuts the size of the list in half; how many
times can we do this?
If we think backwards, in terms of doubling the list, we’ll
need doublings to generate a list of length .
If
, then
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Logarithmic Operators
2-8:
This is used to change bases.
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Logarithm examples
Evaluate
2-9:
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Logarithm examples
, since
By rule 1:
Evaluate
2-10:
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Logarithm examples
Evaluate
2-11:
But many calculators don’t do base 2!
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Logarithm examples
2-12:
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2-13:
Logarithm examples
Logarithms can also be very useful for comparing very
large or small numbers.
For example:
is this true?
Neither number can be calculated directly without risking
overflow.
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2-14:
Logarithm examples
Since the logarithm function is monotonically increasing, if
, then
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Summations
2-15:
Often, in analyzing a program’s performance, we’ll need to
add up the number of times an operation is taken.
This is known as a summation.
Summations are typically written as:
Which is equivalent to
(1)
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2-16:
Summations
Typically, you’ll want to replace a summation with a single
expression, known as a closed-form solution.
Some common summations and their closed-form
solutions:
(3)
(4)
(2)
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2-17:
Summations
(5)
Some common summations and their closed-form
solutions:
(7)
(6)
There are more on page 28 of your text.
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2-18:
Recursion
Recursion is a fundamental problem-solving technique
Involves decomposing a problem into:
A base case that can be solved directly
A recursive step that indicates how to handle more
complex cases.
A common recursive example is factorial:
long factorial(int input)
if (input == 1)
return 1;
else
return input * factorial(input - 1);
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2-19:
Recursion
A more interesting example is the Towers of Hanoi.
It’s hard to write an iterative program to solve this, but the
recursive version is startlingly simple:
void towers(int ndisks, Tower startTower, Tower goalTower, Tower tempT
if (ndisks == 0)
return;
else
towers(ndisks - 1, startTower, tempTower, goalTower);
moveDisk(startTower, goalTower);
towers(ndisks - 1, tempTower, goalTower, startTower);
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2-20:
Mathematical Proof
Often, we’ll find ourself needing to mathematically prove
something, such as that a particular algorithm is optimal,
that a certain set of outcomes is impossible, or that a
solution is guaranteed to be correct.
There are lots of invalid proof techniques:
Proof by assertion
Proof by example (this can be valid for existence
proofs)
Proof by appeal to authority
Proof by obfuscation
etc ...
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2-21:
Mathematical Proof
The three most common valid forms of proof are:
Direct proof (sometimes called a constructive proof)
Indirect proof, or proof by contradiction
Inductive Proof
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2-22:
Direct Proof
With direct proof, you set out to directly show how the
hypothesis holds.
Direct proofs are typically very short and direct.
Example:
Hypothesis: Every odd integer is the difference of two
perfect squares.
(A perfect square is a number , where is an
integer, such as 1,4,9,16,25, ...)
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2-23:
Direct Proof
is an
, where
We can write an odd number as
integer.
Hypothesis: Every odd integer is the difference of two
perfect squares.
. These are both perfect squares.
Direct proof is great, when you can do it.
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2-24:
Proof by Contradiction
Proof by contradiction works as follows:
To show that is true, start by assuming is false.
Use this to derive a logical contradiction.
Since being false results in a contradiction, must
be true.
Note that this assumes all hypotheses are either true or
false.
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Proof by Contradiction Example
2-25:
Hypothesis: There are infinitely many prime numbers.
Proof:
Assume not. Let be the largest prime, and be the
product of all primes from 2, ..., .
Let
=
.
For any prime
,
will have a remainder of
1. (by construction)
Since no prime number divides
, no number less
(except 1) divides
than
Therefore
is prime.
This contradicts our assumption that is the largest
prime; therefore our assumption is false, and our
hypothesis proved.
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2-26:
Inductive Proof
Inductive proofs are very similar to recursion.
You establish a base case that is proved directly
This is followed by an inductive step, which shows how the
hypothesis holds for larger cases.
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2-27:
Inductive Proof
Typically, we’ll want to prove that something holds over a
range of values.
The base case will prove the theorem for the initial
values.
The indictive step will show that, if the theorem holds for
, then it holds for .
This provides the machinery needed to prove the theorem
for any particular value of .
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Inductive Proof Example
.
Previously, we saw that
2-28:
.
Base case: Let
Let’s prove this.
(8)
Inductive step. First, we state the inductive hypothesis
that:
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2-29:
Inductive Proof Example
Assuming this is true, adding the th element yields:
(9)
Therefore, we have proved our theorem.
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2-30:
Summary
Hopefully this material is looking familiar to you.
If not:
Read Chapter 2 closely
Rediscover your Discrete Math text
Come talk to me or the TA in office hours.
You also have a homework to help ’jog your memory.’
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