Intro to computer science module 3

Foundations of Computer Science and Software Engineering

module 3/30

WALID S. SABA

ALBERT EINSTIEN

TO THE TEACHER

TO THE STUDENT

SOCRATES

About this CourseThis is an introductory course that will introduce computer science and software engineering to the novice, and it is also a course that will teach even veteran computer scientists some new concepts, or explain concepts they thought they understood well. Concepts are introduced with hands on experience using object-oriented pseudo code (which can very easily be translated to C#, Java, C++, etc.) and functional programming pseudo code (which can easily be translated into F#, Erlang, Python, Haskell, etc.)

About the AuthorWalid Saba has 20 years of experience in information technology, where he worked at such places as the American Institutes for Research, AT&T Bell Labs, Metlife, Nortel Networks, IBM and Cognos. He has also spent 7 years in academia where he has taught computer science at Carelton University, the New Jersey Institute of Technology, the University of Windsor, and the American University of Beirut. He has published over 30 technical articles, including an award wining paper that he recieved at KI-2008 in Germany. Walid holds a PhD in Computer Science which he obtained from Carleton University in1999

MODULE 3 (1 hour)

MATHEMATICAL PRILIMENARIES – SETS, RELATIONS AND FUNCTIONS

Introduction to Computer Science – Preliminaries

A set is an unordered collection of distinct objects

The following is a set that contains 4 distinct objects (in this case numbers)

Although we have to list the objects in some order, the order is not important at all

Since objects in a set are unordered, the sets A, B and C are equal

It is therefore useful to visualize the sets A, B and C as a Venn diagram

Since objects in a set are unordered, the sets A, B and C are equal

Objects are collected in a set for some purpose.

Objects are collected in a set for some purpose. Here are some examples:

Set of countries that have a permanent seat in the United Nations Security Council (countries with veto power)

Objects are collected in a set for some purpose. Here are some examples:

Set of countries that have a permanent seat in the United Nations Security Council (countries with veto power)

Set of US presidents that died while in office

A set can be defined extensionally, by actually enumerating all the objects

EXAMPLES

The set of prime numbers that are less than 19 (a prime number is a number that is divisible by itself and 1 only)

EXAMPLES

The set of actors that starred in The Untouchables movie

The set of prime numbers that are less than 19 (a prime number is a number that is divisible by itself and 1 only)

A set is often defined by comprehension – that is, by some predicate

A is the set of

A is the set of all objects x

A is the set of all objects x such that

A is the set of all objects x such that the property (or predicate) P is true of x

EXAMPLES

EXAMPLESThe set of all objects x such that x is an even number that is greater or equal to 2 and less than or equal to 8

EXAMPLES

The set of all pairs (x,y) such that x is an even number between 1 and 9 and y is the square of x

EXAMPLES

Set Membership

Objects either belong to (are members of) a set, or they are not.

Set Membership

When an object e belongs to (or is a member of) a set S, we write

Set Membership

If an object e does not belong to a set S, we write

Set Membership

If an object e does not belong to a set S, we write

In advanced computer science curses you will encounter another kind of set membership that is not crisp (yes, or no), but fuzzy!

For example, in defining membership in the set of BIRDS, an eagle might be considered a member with degree 1.0, while a penguin might be considered a member with degree 0.6 only

Fuzzy sets and fuzzy membership are beyond the scope of this course, but reading a high-level introduction to the subject is not a bad idea

Empty Set

What happens if the predicate (or property) that defines membership in the set is not true of any object?

Empty Set

Consider these sets:

Empty Set

Now what members belong to this set?

Empty Set

It turns out that the set is an empty set – i.e., it has no members, since there are no objects that are members of D and also members of E

Empty Set

It turns out that the set is an empty set – i.e., it has no members, since there are no objects that are members of D and also members of E

When a set B is empty we write:

The Universal Set

Unlike the empty set, which has no members, the Universal Set, often denoted by U, is the set that has all objects as members, including itself – its is the set of all objects in the domain of discourse

The Universal Set

Unlike the empty set, which has no members, the Universal Set, often denoted by U, is the set that has all objects as members, including itself – its is the set of all objects in the domain of discourse

Graphically, we usually depict sets and the Universal Set as follows:

Set Operations

Note the figure below.

Set Operations

o C is entirely contained in the set A (we say that C is a subset of A)

Set Operations

o C is entirely contained in the set A (we say that C is a subset of A)o E and B have something in common (their intersection is the set H)

Set Operations

o C is entirely contained in the set A (we say that C is a subset of A)

o if we take the union of A and F, it will also contain all the elements of Co E and B have something in common (their intersection is the set H)

Set Operations

SUBSET

A set R is a subset of another set S if every element of R is also an element of S.

We write

We can state this in first-order logic (FOL) as follows:

Set Operations

SUBSET

A set R is a subset of another set S if every element of R is also an element of S.

We write

What the above says is the following:R is a subset of S if and only if any x that belongs to R also belongs to S

Set Operations

The union of a set R and a set S is a set that has the elements that are in R and those that are in S

We write

Set Operations

The union of a set R and a set S is a set that has the elements that are in R and those that are in S

We write

What the above says is the following:The union of R and S contains any x that is an element of R or an element of S

Set Operations

INTERSECTION

The intersection of A and B is a set that has all the elements that are in both, A and B

We write

Set Operations

What the above says is the following:The intersection of R and S contains any x that is in both R, and S

INTERSECTION

The intersection of A and B is a set that has all the elements that are in both, A and B

We write

Set Operations

COMPLEMENT

The complement of a set A is a set that contains all the elements that are not in A

We write

Set Operations

What the above says is the following:The complement of a set A is a set that has all the elements that are not in A

COMPLEMENT

The complement of a set A is a set that contains all the elements that are not in A

We write

Set Operations

For example, if S = {x,y,z} then P(S) = {{x,y,z},{x,y},{x,z},{y,z},{x},{y},{z},{ }}

If |S| is the size of a set S, then |P(S)| = 2 | S|

The Power Set of a set S, denoted to by P(S), or 2S, is the set of all subsets of S

Correspondence between Set Theory and Logic

Set Theory Explanation of Implication

Let us re-visit the truth table of the implication logical operator, where R => S is false in one case only, namely when R is true and S is false

Recall also that corresponds to

Let us assign truth values for R and S – where T means we are in the set

T T TIt possible to be in R and S at the same time

It possible to be in S but not in R

F T TF F T

It is also possible not to be in R and not to be in S

T T TT F FF T TF F T

But it is not possible to be in R and not in S at the same time

Sets – An Example

Suppose that in a certain school we have the following data:

Suppose the school is holding a Christmas party and we want to make sure we have a tag for everyone in the school. How many tags should we make?

Sets – An Example

Suppose that in a certain school we have the following data:

We need a tag for every student, teacher and staff member. The entire set is

Suppose the school is holding a Christmas party and we want to make sure we have a tag for everyone in the school. How many tags should we make?

Sets – An Example

OPERATIONS

Sets – An Example

OPERATIONS

(1) How would you find out the number of presents the school should prepare if the school is giving a present to every teacher and every staff member?

(2) How can we know the names of those that will receive more than one present?

Relations

Cartesian product is an operation that returns a set which is the product between two or more sets.

CARTESIAN PRODUCT

For example, the Cartesian product between two sets X and Y, denoted as X x Y is the set of all pairs (x,y) where x is from X and y is from Y

Relations

Cartesian product is an operation that returns a set between two or more sets.

CARTESIAN PRODUCT

Formally, we have the following:

Relations

CARTESIAN PRODUCT

Note that, in general,

Relations

CARTESIAN PRODUCT

Note that, in general,

If |X| is the size of X and |Y| is the size of Y then |X x Y| is |X|.|Y|

Relations

A relation between two sets is some subset of the Cartesian product between the two sets

Relations

For example, assuming that N is the set of natural numbers, and S is the set of all possible words in English, then the following is one possible relation that is a subset of S x N

R = { (“Hendrix”, 7), (“Dylan”, 5), (“Marley”, 6), (“Morrison”, 8) }

Relations

This relation might be used to pair strings with their size (number of characters in the string)

For example, assuming that N is the set of natural numbers, and S is the set of all possible words in English, then the following is one possible relation that is a subset of S x N

Visualizing a Relation

Relations and Quantifiers in First-Order Logic

Recall the quantifiers of first-order logic:

This says that for all x, R is true of x (or, R is true of any x)

This says that there exists some x that R is true of (or, R is true of some x)

But what if the property (or predicate) R is a 2-place property, or a property that is true of two variables?

The predicate R here is true of some x and y if x and y are related to each other by R – or, if the pair (x,y) is in the relation that corresponds to R

For example, we can have the following 2-place predicates:

john knows mary

carlos likes pineapple

Ottawa is the capital of Canada

Length of the string “abba” is 4

BMW is made in Germany

Suppose we have a relation Likes that might hold between some x and some y, where x is from some set X and y is from set Y

When we have 2 variables and more than the order of the quantifiers is important

Here there is at least one x that is related to some (at least one) y

Here every x is related to some (at least one) y

Here there is at least one x that is related to every y (at least one person likes all fruit types!)

Here we are saying that every x is related to every y (everyone likes all fruits types). Can you complete the picture of this relation?

Properties of Relations

A relation R can have any of these properties

AN EXAMPLE

The subset relation on sets is

AN EXAMPLE

Why is the subset relation not symmetric?

Consider the relation (‘less than or equal to’) on N x N

Here are some pairs in this relation

= {(0, 1), (0, 2), (0, 3), ... (1, 2), (1, 3), (1, 4), (1, 5), ... (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), ... ... }

Is this relation reflexive? Symmetric? Transitive?What about the equality (=) relation on natural numbers?

Relations in Data Modeling

Consider the following sets, that represent certain data objects:

Several relations can be defined on these sets to represent, for example, data about employees in a certain organization:

These relations might have the following data stored in them:

Company Name Age

Company City

Company Name Age

Company City

Note that a relation on sets is itself a set: a set of pairs, triples, etc.. In mathematics we call these objects tuples. In database systems we call these objects records, and we call the relations tables

Company Name Age

Company City

LOOKING AHEAD

These two relations are connected somehow (they share one data object, namely the company an employee works for)

Given these tables/relations, how would we get the list of employees that work in Tokyo?

Functions

A function is a relation between a set of inputs X and a set of outputs Y, or a map from some domain X to another domain Y

Functions

An example function that maps objects in some set to their color

A function is a relation between a set of inputs X and a set of outputs Y, or a map from some domain X to another domain Y

Functions

012345...

0123456789

1011...

X: input

Y: output

A function f is a relation between a set of inputs X and a set of outputs Y, or a map from some domain X to another domain Y

Functions

012345...

0123456789

1011...

X: input

Y: output

f(0) = 0

Functions

012345...

0123456789

1011...

X: input

Y: output

f(1) = 2

f(0) = 0

Functions

012345...

0123456789

1011...

X: input

Y: output

f(2) = 4f(1) = 2f(0) = 0

Functions

012345...

0123456789

1011...

X: input

Y: output

f(3) = 6f(2) = 4f(1) = 2f(0) = 0

Functions

012345...

0123456789

1011...

X: input

Y: output

f(4) = 8f(3) = 6f(2) = 4f(1) = 2f(0) = 0

Functions

012345...

0123456789

1011...

X: input

Y: output

f(5) = 10f(4) = 8f(3) = 6f(2) = 4f(1) = 2f(0) = 0

Functions

012345...

0123456789

1011...

X: input

Y: output

f(5) = 10f(4) = 8f(3) = 6f(2) = 4f(1) = 2f(0) = 0

Can you guess what this function is doing?

Functions

012345...

0123456789

1011...

X: input

Y: output

f(5) = 10f(4) = 8f(3) = 6f(2) = 4f(1) = 2f(0) = 0

Can you guess what this function is doing?The function seems to map every number to its double ...

Functions

A function is thus a relation from a set of inputs to a set of outputsA function is sometimes also called a map, since it maps inputs to outputs

Functions

In engineering, machine is used as a metaphor for function since it takes some objects as input, does something with them, and produces new objects

Functions

In engineering, machine is used as a metaphor for function since it takes some objects as input, does something with them, and produces new objects

In computer science, a function is a program or a procedure

Functions

The output of a function is some value, and so it can be the input to another function

DoubleSquare3

Functions

DoubleSquare3 9

Functions

DoubleSquare3 9 18

Functions

DoubleSquare5

Functions

DoubleSquare5 25

Functions

DoubleSquare5 25 50

Functions

Functions can have any number of inputs:

Functions

f1 : if (x > y) then x else y

Functions

f2 : (x * x) + (y * y)

Functions

f2 : (x * x) + (y * y)

What do you thing the function f1 does?

Functions

f2 : (x * x) + (y * y)

What do you thing the function f1 does?What about the function f2?

Functions

What if we want to return more than a single output value for the same input?

Functions

For example, suppose we want a function that takes a string and return

1- the reverse of the string,2- the size of the string (number of characters in the string), and3- the original string with no duplicates

Functions

f“santana”

Functions

We can return a single output value, which is a triple that contains the three values

f“santana”

Functions

f“santana” (“anatnas”, 7, “sant”)

Functions

f“santana” (“anatnas”, 7, “sant”)

In some programming languages these tuples (pairs, triples, 4-tuples, etc.) are exactly as you see it here. In other languages, however, we need to define a structure that holds this triple. This is done as follows:

structure Triple{ string reverseOfString, integer numberOfCharactersInString, string stringWithNoDuplicates}

Conditions in Functions

In most cases functions compute an output value in several ways, based on some conditions of the input

For example, the max function, which returns the maximum of two values, uses an if-then-else construct:

max (x,y) = if (x > y) then return x else return y

As another example, recall the truth table of the logical and operator

and (x,y) = if ((x == T) & (y == T)) then return T else return F

A function to compute logical and using if-then-else

and T T = Tand x y = F

In some languages we can handle the different input cases by doing pattern-matching on the input values

and T T = Tand x y = F

In some languages we can handle the different input cases by doing pattern-matching on the input valuesIn these languages cases are tested in the order they appear. If

the input does not match one case, the next case is applied.

1) Try different inputs on this function2) Define the logical OR function in the same manner

Exercises for Modules 1, 2 and 3

1. Devise a coding scheme to represent simple text in binary code (i.e., as a sequence of 1’s and 0’s). You can assume that the text will contain only lower case characters of the English alphabet, the period (‘.‘), the comma (‘,‘) and the space (‘ ‘) character. According to your coding scheme, how many bits would be required to code a piece of text that has 100 characters.

2. Consider the following statements:a) Animals that have wings flyb) Eagles have wingsc) John owns a pet eagle d) The name of the eagle John owns is ‘Clay’

2.1 Represent the above statements in first-order logic (FOL)2.2 Show/prove using logical inefrencing rules that Clay flies

3. Show/prove using the inference rules of logic that the following are true:a) (A U (B \ C)) = (A U B) \ (A U C)

b) If (A µ B) and (B µ A) then A = B

c) if (A U B) = A then (A µ B)

4. Suppose we have an operation called difference (‘ – ‘) such that (A – B) returns a set that has all the elements that are in A but in B. How can you get the intersection of A and B using this operation only?

Introduction to Computer Science

END OF MODULE 3

MATHEMATICAL PRILIMENARIES – RECURSION AND INDUCTIVE STRUCTURES (part a)

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