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Module 2: Language of Mathematics
Theme 1: Sets
A set is a collection of objects. We describe a set by listing all of its elements (if this set is finite and
not too big) or by specifying a property that uniquely identifies it.
Example 1: The set
of all decimal digits is
But to define a set of all even positive integers we write:
where
is a natural number
The last definition can be also written in another form, namely:
is an even natural number
In the rest of this course, we shall either write
property describing
or
property describing
,
where
or
should be read as such as. Both are used in discrete math, however, we prefer the former.
This notation is called the set builder.
Let
be a set such that elements
belong to it. We shall write
if
is an element of
. If
does notbelong to
we denote it as
.
Uppercase letters are usually used to denote sets. Some letters are reserved for often used sets such
as the set of natural numbers
(i.e., set of all counting numbers), the set of integers
(i.e., positive and negative natural numbers together with zero), and
the set of rational numbers which are ratios of integers, that is,
. A
set with no elements is called the empty (or null) set and is denoted as .
The set
is said to be a subset of
if and only if every element of
is also an element of
.
We shall write
to indicate that
is a subset of
.
Example 2: The set is a subset of . Actually, in this case is a proper
subset of
, and we write it as
. By proper we mean that there exits at least one element of
that is notan element of
(in our example such elements are
, or
or
).
Two sets
and
are equal if and only if they have the same elements. We will subsequently
make this statement more precise. For example, the sets
and
are
equal since order does not matter for sets. When the sets are finite and small, one can verify this
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by listing all elements of the sets and comparing them. However, when sets are defined by the set
builder it is sometimes harder to decide whether two sets are equal or not. For example, is the set
(i.e., solutions of this weird looking equation
, where
)
equal to
? Therefore, we introduce another equivalent definition:
if whenever
, then
and whenever , then . The last statement can be written as follows:
if and only if
(1)
(To see this, one can think of two real numbers
and
that are equal; to prove this fact it suffices to
show that
and
.) This equivalence is very useful when proving some theorems regarding
sets.
If
is finite, then the number of elements of
is called its cardinality and denoted as
, that
is,
number of elements in A
A set is said to be infinite if it has an infinite number of elements. For example, the cardinality of
is
, while
is an infinite set.
The set ofall subsets of a given set
is called the power set and denoted
.
Example 3: If
, then there are
subsets of
, namely:
Thus the cardinality of
is
.
Now, we shall prove our first theorem about sets.
Theorem 1. If
, then
.
Proof:1 The set has elements and we can name them any way we want. For example,
. Any subset of
, say
, contains some elements form
. We can list these
elements or better we can associate with every element
an indicator
which is set to be
if
and zero otherwise. More formally, for every subset
of
we construct an indicator
with understanding that
if and only if the
th element of
belongs to
; other-
wise we set
. For example, for
, the identifier of
is
since
is not
an element of
(i.e.,
) while
, that is,
and
. Observe that every set of
has a unique indicator
. Thus counting the number of indicators will give us the
desired cardinality of
. Since
can take only two values, and there are
possibilities the totalnumber of indicators is
, which is the cardinality of
. This completes the proof.
The Cartesian product of two sets
and
, denoted by
, is the set oforderedpairs
where
and
, that is,
and
1This proof can be omitted in the first reading.
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A B
U
A B
A
B
U
A B
A
U
B
U
B
B U B=A B
Figure 1: Venn diagrams for the union, intersection, difference, and complementary set.
Example 4: If and , then
In general, we can consider Cartesian products of three, four, or
sets. If
, then an element
of
is called an
-tuple.We now introduce set operations. Let
and
be two sets. We define the union
, the
intersection
, and the difference
, respectively, as follows:
or
and
and
Example 5: Let and , then
We say that
and
are disjoint if
, that is, there is no element that belongs to both
sets.
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Sometimes we deal with sets that are subsets of a (master) set
. We will call such a set the
universal set or the universe. One defines the complement of the set , denoted as , as
.
We can represent visually the union, intersection, difference, and complementary set using Venn
diagrams as shown in Figure 1, which is self-explanatory.
When and are disjoint, the cardinality of is the sum of cardinalities of and , that
is,
(provided
). This identity is not true when
and
are not disjoint,
since the intersection part would be counted twice!. To avoid this, we must subtract
yielding
The above property is called the principle of inclusion-exclusion. An astute reader may want to
generalize this to three and more sets. For example, consider the sets from Example 5. Note that
, while
,
and
, thus
.
We have already observed some relationships between set operations. For example, if
, then
, but
. There are more to discover. We list these identities
in Table 1.
Table 1: Set Identities
Identity Name
Identity Laws
Domination laws
Idempotent laws
Complementation laws
Commutative laws
Associative laws
Distributive laws
De Morgans laws
We will prove several of these identities, using different methods. We are not yet ready to use
sophisticated proof techniques, but we will be able to use either Venn diagrams or the the principle
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expressed in (1) (i.e., to prove that two sets are equal it suffices to show that one set is a subset of
the other and vice versa). The reader may want to use Venns diagram to verify all the identities of
Table 1.
Example 6: Let us prove one of the identities, say De Morgans law, showing that
and
. First suppose
which implies that
. Hence,
or
(observe this by drawing the Venn diagram or referring to the logical de Morgan laws discussed in
Module 1). Thus,
or
which implies
. This shows that
.
Suppose now that
, that is,
or
. This further implies that
or
.
Hence
, and therefore
. This proves
, and completes the proof
of the De Morgan law.
Exercise 2A: Using the same arguments as above prove the complementation law.
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Theme 2: Relations
In Theme 1 we defined the Cartesian product of two sets
and
, denoted as
, as the set of
ordered pairs
such that
and
. We can use this to define a (binary) relation
.
We say that
is a binary relation from
to
if it is a subset of the Cartesian product
. If
, then we write
and say that
is related to
.
Example 7: Let
and
. Define the relation
as
divides
where by divides we mean with zero remainder. Then
We now define two important sets for a relation, namely, its domain and its range. The domain
of
is defined as
and
for some
while the range of
is the set
and
for some
In words, the domain of
is composed of all
for which there is
such that
. The set of
all
such that there exists
is the range of
.
Example 8: In Example 7 the domain of is the set while the range is . Observe
that domain of
is a subset of
while the range is a subset of
.
There are several important properties that are used to classify relations on sets. Let
be a
relation on the set
. We say that:
is reflexive if
for every
;
is symmetric if
implies
for all
;
is antisymmetric if
and
implies that
;
is transitive if
and
implies
for all
.
Example 9: Consider
and let
. Observe that
the relation
is:
notreflexive since
;
notsymmetric since for example
but
;
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notantisymmetric since
and
but
;
nottransitive since
and
, but
.
On the other hand, consider this relation
. The
relation
is reflexive, transitive, but not symmetric, however, its antisymmetric, as easy to check.
Exercise 2B: Let
. Is the following relation
reflexive, symmetric, antisymmetric, or transitive?
Relations are used in mathematics and in computer science to generalize and make more rigorous
certain commonly acceptable notion. For example,
is a relation that defines equality between
elements.
Example 10: Let
be the set of rational numbers, that is, ratios of integers. Then
means
that
has the same value as
. For example,
. The relation
partitions the set of all
rational numbers
into subsets such every subset contains all numbers that are equal. Observe that
is reflexive, symmetric and transitive. Indeed,
,
is the same as
, and finally if
and
, then
. Such relations are called equivalence relations and they play important
role in mathematics and computer science.
A relation
that is reflexive, symmetric, and transitive is called an equivalence relation on the
set
. As we have seen above, the relation
divides (actually, partitions) the set of all rational
numbers into disjointsets that cover the the whole set of rational numbers. Let us generalize this. For
an equivalence relation
we define the equivalence set for any
denoted by
as follows
In words,
is the set ofall elements such that and are related by . This is a special set,
as we formally explain below. Informally, the two different equivalence sets
and
are disjoint and
the whole set
is partitioned into disjoint equivalence sets (i.e.,
is the sum of disjoint equivalence
sets).
More formally, observe that if
, then
. Indeed, let
. We shall prove that
, hence
. Thus
and from
and transitivity we conclude that
, hence
, as needed. In a similar manner we can prove that
which proves that
.
Clearly if
, then
(by definition of
). The latter can be expressed in a different, but
logically equivalent, manner: If
, then
(this is an example of counterpositive
argument discussed in Module 1: Basic Logic). We should conclude that the set can be partitioned
into disjoint subsets
such that every element of
belongs to exactly one equivalence class
. In
other words, the set
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is a partition of
.
There is one important example of equivalence classes, called congruence class, that we must
discuss.
Example 11: Fix a number
that is a positive integer (i.e., a natural number). Let
be the set of all
integers. We define a relation on by if is divisible by , that is, there is an integer
such that
. We will write
for
, or more often
. Such
a relation is also called congruence modulo n. For example,
since
, but
since
is not divisible by
.
It is not difficult to prove that
is reflexive, symmetric and transitive. Indeed,
since
. If
, then
since
implies
. Finally,
let
and
. That is,
and
. Then
,
hence
.
Since
is an equivalence relation, we can define equivalence classes which are called congruence
classes. From the definition we know that
for some
Example 12: Congruence classes modulo
are
In words, an integer belongs to one of the above classes because when dividing by
the remainder is
either
or
or
or
or
, but nothing more than this.
We have seen before that the relation
was generalized to the equivalence relations. Let us do
the same with well known
relation. Notice that it defines an order among of real number. Let now
if
and
. Clearly, this relation is reflexive and transitive because
and if
and
, then
. It is definitely notsymmetric since
doe snot imply
unless
. Actually, it is easy to see that it is antisymmetric since if
and
, then
. Wecall such relations partial orders. More precisely, a relation
on a set
is partial order if
is
reflexive, antisymmetric, and transitive.
Example 13: Let if divides (evenly) for . This relation is reflexive and transitive as
we saw in Example 11. It is not symmetric (indeed,
divides
but not the other way around). Is it
antisymmetric? Let
divides
and
divides
, that is, there are integers
and
such that
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and
. That is,
, hence
. Since
we must have
.
Thus
is antisymmetric.
A reflexive, antisymmetric, and transitive relation
is called partial order (not just an order or
total order) since for
it may happen that neither
nor
. In Example 13 we see
that
and
. If for all
we have either
or
, then
defines a
total order. For example, the usual ordering of real numbers defines a total ordering, but pairs of real
numbers in a plane define only a partial order.
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Theme 3: Functions
Functions are one of the most important concepts in mathematics. They are also special kinds of
relations. Recall that a relation
from
to
is a subset of the Cartesian product
. Recall
also that the domain of
is the set of
such that there exists
related to
through
, that is,
. For relations it is not important that for every
there is
related to
by
. Moreover, it is
legitimate to have two
s, say
and
such that
and
for some
. These two properties
are eliminated in the definition of a function. More formally, we define a function denoted as
from
to
as a relation from
to
having two additional properties:
1. The domain of
is
;
2. If
and
, then
.
The last item means that if there is an
such that it is related to
and
, then
must be equal to
. In other words, there is no
that has two different values of
related to it.
We shall use lowercase letters
,
,
, etc. to denote functions. Furthermore, when
we shall
write it as
. Finally, we will also use another standard notation for functions, namely:
Functions are also called mappings or transformations.
The second property of the function definition is very important, so we characterize it in another
way. Consider a relation
on
and
. Define
Observe that
is a set. It may be empty, may contain one element or many elements. When
is
a function, then
is not empty for every
and in fact it contains exactly one element that is
called an image of
. More generally, the image of
denoted as
for a function
is
defined as
for some
In other words,
is a subset of
for which there is
such that
. For example in
Figure 2 the image of
is
.
Example 14: (a) Consider the relation
from
to
. It is a function since every
has exactly one image in
. In fact,
,
and
. Figure 2 shows a graphical representation of this function.
(b) The relation
is not a function since
and
have the same
image
.
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1
2
3
a
b
c
d
Figure 2: The function
defined in Example 14.
x
321-1-2-3
8
6
4
2
x
321-1-2-3
8
6
4
2
(a) (b)
Figure 3: Plots of two functions: (a)
; (b)
.
Functions are often represented by mathematical formulas. For example, we can write
for every real
, or more formally
or
To visualize such functions we often graph them in the
coordinates where
.
Example 15: In Figure 3 we draw the functions
and
. Both functions
are defined on the set of reals
which is the domain for both functions. Since
and
hence the range of
is the set of nonnegative reals while for
it is the set of positive reals. That is,
and
.
Exercise 2C: What is the image of
for
? What about the image of
over the
function
?
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There are some functions occurring so often in computer science that we must briefly discuss
them here. The first function, the modulus operator, we already studied in Example 11. We say
that
is equal to the remainder when
is divided by
(we, of course, implicitly assume that
, i.e.,
and
are integers). We recall that
is equivalent to
used
before. For example, and .
We shall write this function as
with the understanding that
is the remainder of the division
. The domain of such a function is
the set of integers, while the image (or range) is the set of natural numbers. In fact, we can restrict the
range of
to the set
because the remainder of any division by
must be an integer
between
and
.
The other important and often used functions are the floor and ceiling of a real number. Let
, then
the greatest integer less than or equal to
the least integer greater than or equal to
For example,
Finally, we introduce some classes of functions as we did with relations. Consider the function
shown in Figure 3(a). We have
, that is, there are two values of
that
are mapped into the same value of
(or with the same image). This is an example of a function that is
not one-to-one or injective. We say that a function
from
to
is one-to-one or injective if there
are
such that if
, then
. In other words, for one-to-one function
for each
there is at most one
with
. The function in Figure 3(b) is one-to-one,
as easy to see.
Example 16: Consider
from
to
. This function is injective.
How to know weather a function is one-to-ne or not? We provide some conditions below. We
first introduce increasing and decreasing functions. A function
is increasing (non-
decreasing) if ( ) whenever for all . For example, the
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function
is increasing in the domain
(cf. Figure 3(b)). Similarly, a function
is decreasing (non-increasing) if (
) whenever for all
. For
an increasing (decreasing) function the bigger the value of
is, the bigger (smaller) the value of
will be.
The function
plotted in Figure 3(a) is neither increasing or decreasing in the domain
. However, it is a decreasing function for all negative reals and increasing in the set of all positive
reals.
In Example 16 we have
. A function
from
to
such that
is said to be onto
or surjective function.
Example 17: Let
be such that
, where
is the set of positive real
numbers. Clearly,
, thus it is onto
. But if we define
with
, then such a function is not surjective.
A function
that is both injective and surjective is called a bijection. The function in
Example 16 is a bijection while the function in Example 17 is not. For a bijection we can define an
inverse function as
that is,
and
switch their roles. Observe that we do not need to have bijection in order to define
the inverse since: (i) the domain of the inverse function is
and by the definition of a function, for
every
there must be
such that
; (ii) There must be only one
such that
and this is guaranteed by the requirement that
is one-to-one function.
Example 18. Let
be such that
. This is not a one-to-one function. Let us
restrict the domain
to the set of nonnegative reals
and we do the same with the
range, that is,
. Now
has an inverse function defined on
which
is
.
Finally, we define the composition of two functions. Let
and
Then for every
we find
, but for such
we compute
. The resulting
function is called the composition of and and is denoted as .
Example 19: Let
Then
Example 20: Let
and
. The composition
.
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Theme 4: Sequences, Sums, and Products
Sequences are special functions whose domain is the set of natural numbers
or
, that is,
is a sequence. We shall write
to denote an
element of a sequence, where the letter
in
can be replaced by any other letter, say
or
.Since a sequence
is a set we often write it as
or simply
.
Example 21: Let
for
. That is, the sequence
starts with
If
, then the sequence begins with
Finally,
looks like
We can create another sequence from a given sequence
by selecting only some terms. For
example, we can take very second term of the sequence
, that is,
. This amounts
to restricting the domain to a subset of natural numbers. If we denote this subset as
, then we
can denote such a sequence (subsequence) as
. Another way of denoting a subsequence is
where
is a subsequence of natural numbers, that is,
. It is usually
required that
, that is,
is an increasing sequence.There is one important subsequence that we often use. Namely, define
.
Sometimes, we shall denote such a sequence as
.
Example 22: Let
. The first terms are
. Take every second term to produce
a sequence that starts
. We can write it as
or as
.
Sequences are important since they are very often used in computer science. They are frequently
used in sums and products that we discuss next. Consider a (sub)sequence
and add all the elements to yield
To avoids the dots
we have a short hand notation for such sums, namely
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In the above, we use different indices of summation
,
or
since they do not matter. What matters
is the lower bound
and the upper bound
of the index of summation, and the sequence
itself.
In a similar manner we can define the product notation. For the above case instead of writing
we simply write
Example 23: Here are some examples:
Exercise 2D: Find
1.
.
2.
.
We have to learn how to manipulate sums and products. Observe that
In the above we change the index of summation from
to
. We obtained exactly the same
sum
. In general, when we change index
to, say
for some
, we must change the lower summation index from
to
, the upper
summation index from
to
, and
must be replaced by
.
Example 24: This is the most sophisticated example in this module, however, it is important that the
reader understands it. We consider a special sequence called the geometric progression. It is defined
as follows: Fix
and define
for
. This sequence begins with
Let us now consider the sum of the first
terms of such a sequence, that is,
(2)
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Can we find a simple formula for such a sum? Consider the following chain of implications
where the second line follows from the change of the index summation
, in the third line we
factor
in front of the sum, while in the last line we replaced
by
as defined in (2). Thus
we prove that
from which we find
:
as long as
. Therefore, the complicated sum as in (2) has a very simple closed-form solution
given above. An unconvinced reader may want to verify on some numerical examples that these two
formulas give the same numerical value.
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