Set 1. Theory
The first mathematical topic that we have to look at in order to understand how search
engines work is the topic of s ets . We need this because the first step that any search
engine does in trying to answer your question, is to identify a colle ction of web pages
that in some way seem to match your question. That collection is a s et. The way that the
engine finds that set involves certain basic mathematical operations that can be
performed on sets. So here we go.
Definition: A set is a collection of objects.
2
Examples of collections of objects:
My cat, my dog, and my elephant.
All cats.
The animal types {cat, dog, cow, tiger}
All students in ITI 111
The students {John, Maria, Pedro, Paul}.
Students whose names are John, Maria, Pedro and Paul.
3
There are two important ways to specify a set, by a list, and by a rule.
The first way to specify a set is by listing all of the objects that are in it. This is done, on
a page, by using curly braces to represent the beginning and end of the set and listing the
things that are in the set, or their names, usually separated by commas. So we might
have a set that contains the letters a,b,c,d,e,f,w. We could give that set a name (C) and
write in equations something like C={a,b,c,d,e,f,w}.
4
The second way of specifying a set: giving a rule that tells what belongs to the set. The
way of writing that rule also uses curly braces, but it uses the colon punctuation mark to
stand for the words such that. So if we write something like:
B {letters : theletteris a vowel} , then we have given an expression that means the set
of letters that are vowels. You would read the above as a set of letters such that the
letter is a vowel.
HOW ELSE CAN WE WRITE THIS SET?
B= {a, e, i, o, u}
Its important to note that the order in which the e lements of a set are listed does not
matter. The set containing the letters a and t is exactly the same set as the set containing
the letters t and a. So {a, t} {t , a}
Every element in the set is unique. So {a, a, t, t, t} = {a, t}.
Membership notations: x in X.
In addition quite sensibly, we say that t wo sets are the same if and only if their unique
elements are exactly the same (regardless of order). This is the definition of
equality of sets.
5
Notice that I have snuck in a new mathematical word, element here without even
announcing it.
If you ever studied chemistry you learned that the word element means something like
Hydrogen or Helium or Carbon.
In mathematical language we use the word ele ment as another way of talking about
the things that belong to a set. Its not the only word we will use. Sometimes we will use
the word member and we will say that for example a is a member of the set {a, t} .
There are other ways to say it also. a belong to the set or the set contains a.
NOTATION: a belongs to A.
The membership symbol looks like . This looks a little bit like the Greek letter epsilon
but its actually a specific mathematical symbo l which is a kind of stylized form of the
letter e. (Standing for element of course). Mathematically if we want to say that x is
an element of the set A we simply write x A.
6
1.1. Relations and Operations on Sets
As soon as you have the idea of a set you start asking yourself well what if there is
more than one set? Mathematics approaches questions like that by defining relations
between sets and defining operations that can be performed on sets. Well start with the
operations first.
7
1.1.1. The union of sets:
The first operation that can be performed on sets is to form a new set, which is called the
union of the two sets.
The union of two sets is exactly the set containing all the elements that are in either one
of them, or in both of them.
Of course since this is a rule it can actually be written out using the set theoretic
notation.
The union is represented by a symbol that looks a lot like the capital letter U. It is
sometimes called cup.
A B {x : x A or x B} (well talk more about specifying rules this way later on).
This equation would be said like this: The union of set A and B is equal to the set x
such that x is an element of set A or x is an element of set B
8
SIDE NOTE: We are going to use the word or in a rather strict sense, as it is defined
in logic and mathematics. This is a little bit different from the way the word is used in
everyday English.
For example, in everyday English we might say something like either the te acher will
show up, or he wont. When we use the word that way of course we mean that its
going to be one of them or the other, but obviously it cant be both.
9
If we describe several different sets by listing their elements, then we will probably
wind up listing the elements in their union.
{a,b,4}{a, c,3,11 {c, b, a,3,11,4}
}
CLASS EXCERSIZES
{a, b, tigers, 111} {tigers, 111, b, a}={a,b,111,tigers}
{a, b} {b, c} {c, d}={a,b,c,d}
{letter: the letter is a vowl} {letter: the letter is a consonant}={all letters}
10
1.1.2. The intersection of sets:
The next important operation that can be performed on sets, and it is a really important
one for search engines, is called forming the intersection. The intersection of two sets,
A and B, is the set of all the elements that belong to A and also belong to B.
A B {x : x A and x B}
You may have seen in high school a kind of picture called a Venn diagr am, which
shows this as an overlap.
Intersection
11
The Empty Set
A set with no members/elements. Symbol:
Listing the members: A = {}
But can also be by rule that is never satisfied:
{w: a word that starts with the letters xgfs}
{a: a Math major student in 111}
{p : a female US president}
Note that many of these are intersections!
{all females} INTERSECT {all US presidents}
Exercise: Come up with 2 more intersections such as they results in an empty set.
12
1.2. Relations between sets
We said a little while ago that mathematicians talk about operations on sets and also
about relations between sets.
Mentioned equality (the same elements)
Two other very important relations between sets exist. One of them is called
inclusion, and the other is called complementarity.
13
1.2.1. The inclusion relation:
We say that the set A is included in the set B if every element of A is also an element of
B.
For example the set of all the students in 111 is included in the set of all Rutgers
students. This is because everyone who is taking 111 is also a Rutgers student.
A B if x A implies that x B
More than one way to say this:
A is included in B.
The set A is a subset of the set B.
Of course, you can ask questions like is a set S a subset of itself? By applying the rule
we see that the answer is yes.
One set is a subset of another, if an element that is in the first set means that the element
is also in the second set. So we would have to check: Is it true that if x is an element of
S, then x is an element of S?
Doh.
The answer is obviously. So we can say that every set is included in itself. A A ,
for any set A.
In this example we could say that A is contained in B
14
2. Specifying a set by giving a Rule
For now, lets think about specifying a set by giving a rule.
In order to do this we must first understand integer.
For many centuries the concept of integer was synonymous with the concept of number.
What was meant by integers was the numbers that are used to count things: 1, 2, and so
forth. It was an important achievement in mathematics to recognize that zero is also a
number, but not clear if you can call it an integer.
When we use the word integer were going to mean the counting numb ers: 1,2, and so
forth, but not zero.
Now we can give examples of some large (but familiar) sets that are defined by rules.
For example:
Q N6 {x : x 6n, wherenis an integer} .
This is an interesting kind of definition:
For one thing, it has not only the variable x , which stands for any member of the set,
but it also has another variable in it: n, which appears in the definition itself.
What this means in words is that the set Q contains exactly those numbers which are 6
times some integer.
How many elements are there in the set Q? There are infinitely many. There is one for
every integer.
Are the elements of the set Q integers? Yes, they are. When you multiply an integer by 6
you still get another integer.
Use the letter N1 to represent the integers (were using N1 , to represent this as we start
counting at one rather than at zero.)
Lets pull together some of the things weve said about our strange set Q. We recall that
its elements are exactly 6 times an integer. And weve noticed that it is also a set of
integers. Mathematically we can write: Q {x : x 6n for n N1}
Since these numbers are all integers, we could write the mathematical expression that which
says Q is a subset of N1 .
Q N1
Now how many elements does the set Q have? Even though it is a subset of the integers,
it has the same number of elements as the set N1 . This is easy to see because for every
integer n we have a corresponding x 6n that is an element of Q.
On the other hand if we just start listing the numbers in a row 1,2,3,4,5, 6,7,8,9,,
12,13,, and putting a circle around the ones that are in the set Q, we see that only one out
of 6 of them is in the set Q.
This is whacky! We have two sets that have the same number of elements in them and
yet its perfectly clear that one of them is 6 times as big as the other. This is one of the
so- called paradoxes of the infinite .
DIGRESSION: Baruch Spinoza, the Jewish Dutch philosopher was expelled from the
synagogue; the reasons (indeed the historicity of this statement) are not fully clear. But
by one account his offense involved claiming that God could not exist because if God
does exist he is infinite, but nothing can be infinite, because if something extend to
infinity in both directions we could cut it in the middle, and each piece would be infinite.
But that would mean that half of it was equal to the whole thing. And that was logically
impossible. This was not resolved until the 19 th Century, when Georg Cantor showed
t hat there could be a mathematically consistent theory describing infinite sets.
Russells Paradox: ready to have your mind blown??
Is Set theory complete? No. Why, because it is possible to define a set and to show
that the set we have defined cannot possibly exist.
How can we define a set that cannot possibly exist?
Sets can be members of sets.
A = set of all sets of 2 consecutive integers: WRITE IT OUT. Infinite!
S = set of all sets of students in Rutgers Spring 11 classes WRITE IT OUT
B = set of all sets with 5 elements. Infinite!
D = set of all sets that are not members of themselves.
If D is NOT an element of the set D, then D is a set which is not a member of itself;
therefore, it should be an element of the set D.
If D is an element of the set D, then D is NOT a set which is not a member of itself: it is
a member of itself. Were stuck! That is a contradiction. The set D cannot exist. More
on this later.
15
Now lets consider another set, R which is defined
as R { y : y 3n for an integer (1, 2,3...)} . This is also infinitely large,
We can ask the question is the set R contained in the set Q?
It turns out that its not and its pretty easy to prove that its not. The method of proving
it is called counterexample is the simplest (but sometimes the hardest to find) example
of a mathematical proof.
We will show that the set R is not contained in the set Q. Need to show that if an
element is in R, it does not follow that the element is in Q.
To do this, we consider the number 9. The number 9 is clearly in the set R because its 3
times 3. But it is also not in the set Q because it is not 6 times any integer. So it would
be wrong to claim that R is contained in Q. Q.E.D.
Note: Q.E.D. is an acronym for the Latin expression quod erat demonstratum which
means what was to be proved. It is traditional to u se those letters at the end of a
proof.
16
But inclusion might go either way (or not at all). It turns out to be true that the set Q is
contained in the set R. Since its true, and its mathematics we ought to be able to prove
it.
The way that we prove it, as shown on the equations on the facing page, is to note that
any number that is 6 times an integer can also be written as 3 times 2 times an integer.
But 2 times an integer is just some other integer. And so any number that is 6 times
some integer is also 3 times some other integer.
ON A SLIDE:
x 6n
x 3 2n
x 3m, for m 2n
Class exercise: write two sets, by rules, where one contains the other but not vice versa.
NOTES 2
Specifying sets by rules:
{x : x > 1 and x < 5 and x in N1} = {2, 3, 4}.
Explanation: Because the numbers 2, 3, and 4 are greater than 1 and less than 5.
{u : u < 1 and u > 5 and u in N1} = {}
Explanation: Because there is no intersection between numbers less than 1 and numbers
greater than 5.
X={r:
< 4, r in N1}={r in N1: r<16}
Explanation: Because you would square both sides of the equation, which would give
you r<16.
{p: p is a web page and p contains the word dog}=296,000,000 (according to google)
Explanation: Because there are billions of web pages, it is likely that many of them have
the word dog because dog is a commonly used word. I typed dog into google to
get my approximate answer.
17
One more rule:
{A: A is a set with one element}
Whats going on here? Set of sets.
Like any other object, sets can be elements of other sets. Like all the different sets of
photos you have ever taken on vacation: the collection of all of these will have (say) 10
elements; each element is a set of photos that may contain many others.
So is it possible to have a set that contains ALL the other sets? Something like a
universal set that represents infinity? Some set U that satisfies: U A A ' ? It seems
like there should be such a set, since a set can contain anything, even other sets. It seems
like it would be possible to imagine a set that contains every set, and since it seems
possible, mathematicians simply assumed it was for hundreds of years. But, in fact, it is
not.
Because theres a paradox! (get ready to have your mind blown)
Russell Paradox (copied from 13 above) after Bertrand Russell
A = set of all sets of 2 consecutive integers: Infinite!
S = set of all sets of students in Rutgers Spring 11 classes
B = set of all sets with 5 elements. Infinite!
D = Set of all sets that are not members of themselves.
Is Set theory complete? No.
Can we define a set that cannot possibly exist? D is one!
If D is NOT an element of the set D, then D is a set which is not a member of itself;
therefore, it should be an element of the set D.
If D is an element of the set D, then D is NOT a set which is not a member of itself: it is
a member of itself. Were stuck! That is a contradiction. The set D cannot exist. And
since we have at least one set that cannot exist, we have demonstrated that it is not
possible for every set to exist, and therefore it is not possible to construct a set that
contains every set. Mathematicians say that set theory is therefore incomplete.
It would seem like this is a problem, and for many years this bothered people. But then
Kurt Gdel demonstrated that every type of basic axiomatic (simple rule- based)
mathematics must be incomplete. Even as in the specific case Gdel focused on
basic arithmetic! Gdels Incompleteness Theorem, as it came to be known, is one of
the most important discoveries in 20th Century mathematics, and it came about because
of these confusing notions of infinity and completeness in set theory.
18
3. Sizes of sets
Were going to use the expression |A| to mean the size of the set A.
Examples of sizes of sets:
|{a, b, Rutgers}| 3
|{Canada, Mexico}| 2
| set of all 111 students | = about 65
| set of 111 students set of female RU students | = about 20
| set of 111 students set of 111 instructors| = students + 2
Now if the set A is contained in the set B, can it be that the size of A is bigger than the
size of B?
No: every element of A must be an element of B. But if we match the elements of B
up to the elements of A, there will be at least one element of A left over. QED
Class exercise, sizes for union and intersection:
Three cases:
A contained in B
A overlaps B
A does not intersect B
A has 5 elements, B has 10. How many elements in C=A union B and D = A intersect B
in each case above? Can you tell exactly, or approximately?
Class exercise, sizes for union and intersection:
Three cases:
A contained in B
A overlaps B
A does not intersect B
A has 5 elements, B has 10. How many elements in C=A union B and D = A intersect B
in each case above? Can you tell exactly, or approximately?
1. A contained in B
=> C has 10, D has 5.
2. A intersects B
=> C has at least 10, D had at least 1
3.A does not intersect B:
=>C has 15, D has 0.
Other way to ask:
Given the sizes of x and y - ------ >
|A|=3 and |B|=6
What is the smallest A B?
A={1,2,3} and B={9,4,5,6,7,8}
So the smallest intersection is 0
What is the biggest the intersection can be?
The biggest intersection is 3 because |A|=3
0 {A B}minimum of |A|, |B|
What is the biggest union? |A B|
The biggest |A B| can be is 9
Remember that |A B| is the set that contains both.
Max {|A|, |B|} |A B| |A|+ |B|
Smallest union can be is max of set sizes and the largest it can be is the sum of
the two sizes.
In fact, |A B| = |A|+ |B| - |A B| : dont count anything twice.
We will look at this more carefully next class.