Language of Mathematics

8/3/2019 Language of Mathematics

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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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