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Data Structures and Algorithms

Discrete Math Review

Discrete Math review

- Logarithmic Functions
- Sets
- Logic
- Induction
- Counting

Sets

- A set is a collection of individual elements in

the domain D. The universal set U contains

every element in D . The null set ? contains

no element. - If A is a set in the domain D , A must be a

subset of the universal set U , denoted as A ?

U. - If A consists of some but not all elements, A

is then called a proper subset of U , denoted

as - A ? U .

Sets

- A set is a collection of definite and separate

objects. - The cardinality of a set is the number of

elements in the set. - A subset of a set is a set comprised of a

sub-collection of the elements of the original

set. - Example Consider the set S 2, 5, 7. The

subsets of S are the sets 2 , 5, 7, 2,5 ,

2,7 , 5,7 , 2, 5, 7and (the empty set). - Fact If a set has cardinality n , then the

number of subsets 2n .

Sets

- A proper subset of a set is a set comprised of a

sub-collection of some, but not all, of the

elements of the original set. - Example Consider the set S 2, 5, 7. The

proper subsets of S are the sets 2 , 5, 7,

2,5 , 2,7 , 5,7 and (the empty set). - S cardinality of 3 ? 23 ? 7 ?

Sets

- There are only two different types of sets
- 1. An Infinite Set is a set that can be placed in

a one-to-one correspondence to a proper subset of

itself. - The Basic Infinite Set is the set of positive

whole numbers 1, 2, 3, 4, 5,. New Notation

The Basic Infinite Set has cardinality . - A Finite Set is a set that cannot be placed in a

one-to-one correspondence to a proper subset of

itself. - Something to think about.2 subsets ?

Logarithmic Functions

- Logarithms were invented about 1614 by John

Napier. The word logarithm was derived from two

Greek words, logos, which means "ratio," and

arithmos, which means "number." - Before calculators and computers were available,

common logarithms were used to do certain kinds

of calculations - Growth and decay
- Big O notation growth of computational complexity

Logarithmic Functions

- Logarithms are exponents
- A logarithmic function is the inverse of an

exponential function - One way to describe a logarithmic function is to

interchange variables in the equation y ax - Thus, x ay is logarithmic

Logarithmic Functions

- For logarithmic functions we use the notation

loga x - Therefore, The following are equivalent.
- x ay and
- y loga x

Logarithmic Functions

- Definition of Logarithm
- Suppose bgt 0 and . For n gt 0, there is a number

p such that logb n p if and only if bp n. - log2 256 ? ? 2p 256
- log10 1000 ? ? 10p 1000
- log7 2401 ? ? 7p 2401

Logic

- Examples Boolean functions NOT, AND, OR, XOR, .

. . - a NOT( a)
- 0 1
- 1 0

Logic

Examples Boolean functions NOT, AND, OR

Logic

- Examples Boolean functions XOR, XNOR, NOR

Logic

Boolean Simplification

Express F in terms of A and B

F AB AB

Using the uniting theorem

? A ( B B ) F

Series

- Arithmetic
- One common type of series is the arithmetic

series (also called an arithmetic progression).

Each new term in an arithmetic series is the

previous term plus a given number. For example

this is an arithmetic series 1471013.... - In this case each term is the previous term plus

3. The difference between each term (the 3 in

this case) is called the "common difference" and

is generally denoted by the letter d. - There are many arithmetic series that have d3.

To specify which series we mean, we need to know

one more piece of information the value of the

first term (usually called "a").

Series

- Arithmetic
- If we're given a and d, then, that specifies a

unique arithmetic series. All arithmetic series

therefore have the following form

a(ad)(a2d)(a3d).... - So in general we can say that the value of the

nth term is a(n-1)d

Series

- Geometric
- Another common type of series is the geometric

series (also called a geometric progression). In

this case, each term is the previous term

multiplied by a given number. - Here's a geometric series, for example
- 261854...
- In this case, each term is the previous term

multiplied by 3. The number you multiply by (in

this case 3) is called the "common ratio" and is

generally denoted by r. - There are many geometric series that have r3. To

specify which series we mean, again we need to

know one more piece of information the value of

the first term (usually called "a").

Series

- Geometric
- A geometric series is uniquely specified by the

values of a and r. Every geometric series has the

following form - aarar2ar3.......
- Writing the series in this way we can see the

formula for working out the value of any term.

The first term is just a. The second term is a

multiplied by r once. The third term is a

multiplied by r twice, and so on. So the nth term

must be a multiplied by r (n-1) times, that's

arn-1.

Series

- Are there others?

The P word.Induction

- The Principle of Mathematical Induction
- Suppose we have an assertion P(n) about the

positive integers. - Then if we show both of (i) and (ii) below, then

P(n) is true for all n gt 1. - (i). P(1) is true
- (ii). For each k gt 1 If P(k) is true, then

P(k1) is true.

The P word.Induction

Prove that 123 n

- Prove that 135 (2n 1) n2

Counting

- Product Rule n ways to do m tasks nm ways.
- How many different bit strings are there of

length 7?

Counting

- Pigenhole Principle If N objects are placed into

k boxes, then there is at least one box

containing at least N/k objects. - Ten persons were born on the 1st, 11th and the

27th of the months of May, August and November.

How many share the exact same birthday? - Tells us nothing of the date. Asserts only that

one exists!

Counting

- Permutations and Combinations
- Permutations Given that position is important,

if one has 4 different objects (e.g. A, B, C and

D), how many unique ways can they be placed in 3

positions (e.g. ABD, ACD, BDA, DBA, BAD, ADB,

ADC, DCA, DAC, CDA etc.) - An ordered arrandement of r elements of a set is

called a r-permutation.

Counting

- Permutations and Combinations
- Combinations If one has 4 different objects

(e.g. A, B, C and D) how many ways can they be

grouped as 2 objects when position does not

matter (e.g. AB, AC, AD are correct but DA is not

ok as is equal to AD.) - An r-combination is simply a subset of the set

with r elements.

Base Conversion

- The base value of a number system is the number

of different values the set has before repeating

itself. For example, decimal has a base of ten

values, 0 to 9. - Binary 2 (0, 1)
- Octal 8 (0 - 7)
- Decimal 10 (0 - 9)
- Duodecimal 12 (used for some purposes by the

Romans) - Hexadecimal 16 (0 - 9, A-F)
- Vigesimal 20 (used by the Mayans)
- Sexagesimal 60 (used by the Babylonians)

Base Conversion

- Successive Division
- 3710 base 6
- 37/6 6 r 1
- 6/6 1 r 0
- 1/6 0 r 1
- ?3710 base 6 101

Base Conversion

- 1016 base 10
- 1x600x611x62 1036 37
- 5810base4
- 11110base13

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