# Mathematics in the Modern World

```Speaking Mathematically
Speaking Mathematically
Remilou Liguarda, Ph.D.
Speaking Mathematically
Overview
1
Variables
2
The Language of Sets
3
The Language of Relations and Functions
Speaking Mathematically
Variables
Definition 1
A variable is a symbol representing a quantity whose value is unknown.
Generally, we use a single letter to represent a variable.
Speaking Mathematically
Example 2
Is there a number with a property that: doubling it and adding 3 gives
the same result as squaring it?
Speaking Mathematically
Example 2
Is there a number with a property that: doubling it and adding 3 gives
the same result as squaring it?
In this sentence you can introduce a variable to replace the potentially
ambiguous word ”it”:
Is there a number x with the property that 2x + 3 = x 2 ?
Speaking Mathematically
Example 3
No matter what number might be chosen, if it is greater than 2 then its
square is greater than 4.
Speaking Mathematically
Example 3
No matter what number might be chosen, if it is greater than 2 then its
square is greater than 4.
Mathematical translation:
No matter what number n might be chosen, if n &gt; 2,then n2 &gt; 4.
Speaking Mathematically
Example 4
Given any two real numbers, there is a real number in between.
Speaking Mathematically
Example 4
Given any two real numbers, there is a real number in between.
Mathematical translation:
Given any two real numbers a and b, there is a real number c such that
a &lt; c &lt; b.
Speaking Mathematically
Example 5
Use variables to rewrite the following sentences more formally.
1 Are there numbers with the property that the sum of their squares
equals the square of their sum?
2
Given any real number, its square is nonnegative.
Speaking Mathematically
Example 5
Use variables to rewrite the following sentences more formally.
1 Are there numbers with the property that the sum of their squares
equals the square of their sum?
2
Given any real number, its square is nonnegative.
Solution:
1
2
Are there numbers a and b with the property that
a2 + b 2 = (a + b)2 ?
Or : Are there numbers a and b such that a2 + b 2 = (a + b)2 ?
Or : Do there exist any number a and b such that
a2 + b 2 = (a + b)2 ?
Given any real number r , r 2 is nonnegative.
Or : For any real number, r , r 2 ≥ 0.
Or : For all real numbers r , r 2 ≥ 0.
Speaking Mathematically
Some Important Kinds of Mathematical Statements
1
Universal Statements
2
Conditional Statements
3
Existential Statements
Speaking Mathematically
Definition 6
A universal statement is a mathematical statement that is true for all
elements in a set. It contains universal quantifiers: all, every, each.
Example 7
1
All positive numbers are greater than zero.
2
The square of every real number is nonnegative.
Speaking Mathematically
Definition 8
A conditional statement is an if-then statement, that is, if one thing is
true, then some other thing is also true.
Example 9
1
If 180 is divisible by 45, then 180 is divisible by 9.
2
If you are healthy, then you are normal.
Speaking Mathematically
Definition 10
An existential statement is a statement that is true if there is at least one
element in a set for which the statement is true.
Example 11
1
There is a prime number that is even.
2
There is a positive number that its square is equal to itself.
Speaking Mathematically
Definition 12
An universal conditional statement is a statement that is both universal
and conditional.
Example 13
1
For all animals a, if a is a dog, then a is a mammal.
2
For all numbers x, if x is greater than 1, then x 3 is greater than x 2 .
Speaking Mathematically
One of the most important facts about universal conditional statements
is that they can be rewritten in ways that make them appear to be purely
universal or purely conditional.
Example 14
1
2
3
4
If a is a dog, then a is a mammal. (purely conditional)
Or : If an animal is a dog, then the animal is a mammal.
If a number x is greater than 1, then x 3 is greater than x 2 . (purely
conditional)
Or : If x is greater than 1, then x 3 is greater than x 2 .
For all dogs a, a is a mammal. (purely universal)
Or : All dogs are mammals.
For all numbers x greater than 1, x 3 is greater than x 2 . (purely
universal)
Speaking Mathematically
Example 15. Rewriting a Universal Conditional Statement
Fill in the blanks to rewrite the following statement:
For all object J, if J is a square, then J has four sides.
1
All squares
.
2
Every square
3
If J
4
If an object is a square, then it
5
For all squares J,
.
, then J
.
.
.
Speaking Mathematically
Example 15. Rewriting a Universal Conditional Statement
Fill in the blanks to rewrite the following statement:
For all object J, if J is a square, then J has four sides.
1
All squares
.
2
Every square
3
If J
4
If an object is a square, then it
5
For all squares J,
.
, then J
Solution:
1
have four sides
2
has four sides
3
is a square; has four sides
4
has four sides
5
J has four sides.
.
.
.
Speaking Mathematically
Example 16. Rewriting a Universal Conditional Statement
Fill in the blanks to rewrite the following statement:
For all real numbers x, if x is nonzero then x 2 is positive.
1
If a real number is nonzero, then its square is
2
For all nonzero real number x,
3
If x
4
The square of any nonzero real number is
5
All nonzero real numbers have
, then
.
.
.
.
.
Speaking Mathematically
Example 16. Rewriting a Universal Conditional Statement
Fill in the blanks to rewrite the following statement:
For all real numbers x, if x is nonzero then x 2 is positive.
1
If a real number is nonzero, then its square is
2
For all nonzero real number x,
3
If x
4
The square of any nonzero real number is
5
All nonzero real numbers have
, then
.
.
.
Solution:
1
positive
2
x 2 is positive
3
is a nonzero real number; x 2 is positive
4
positive
5
positive squares (or : squares that are positive)
.
.
Speaking Mathematically
Definition 17
An universal existential statement is a statement that is universal
because its first part says that a certain property is true for all objects of
a given type, and it is existential because its second part asserts the
existence of something.
Example 18
1
Every real number has an additive inverse.
Or : All real numbers have additive inverse.
Or : For all real numbers r , there is an additive inverse for r .
Or : For all real numbers r , there is a real number s such that s is
an additive inverse for r .
Speaking Mathematically
Example 19. Rewriting a Universal Existential Statement
Fill in the blanks to rewrite the following statement:
Every negative integer has a positive square.
1
All negative integers
.
2
For all negative integers x, there is
3
For all negative integers x, there is an integer z such that
.
.
Speaking Mathematically
Example 19. Rewriting a Universal Existential Statement
Fill in the blanks to rewrite the following statement:
Every negative integer has a positive square.
1
All negative integers
2
For all negative integers x, there is
3
For all negative integers x, there is an integer z such that
.
Solution:
1
have positive squares
2
a positive square for x
3
z is the positive square for x
.
.
Speaking Mathematically
Definition 20
An existential universal statement is a statement that is existential
because its first part asserts that a certain object exists and is universal
because its second part says that the object satisfies a certain property
for all things of a certain kind.
Example 21
1
There is a positive integer that is less than or equal to every positive
integer.
Or : Some positive integer is less than or equal to every positive
integer.
Or : There is a positive integer m that is less than or equal to every
positive integer.
Or : There is a positive integer m such that every positive integer is
greater than or equal to m.
Or : There is a positive integer m with the property that for all
positive integers n, m ≤ n.
Speaking Mathematically
Example 22. Rewriting a Universal Existential Statement
Fill in the blanks to rewrite the following statement:
There is a real number whose product with every real number equals zero.
1
Some
has the property that its
.
2
There is a real number y such that the product of y
.
3
There is a real number y with a property that for every real number
w,
.
Speaking Mathematically
Example 22. Rewriting a Universal Existential Statement
Fill in the blanks to rewrite the following statement:
There is a real number whose product with every real number equals zero.
1
Some
has the property that its
2
There is a real number y such that the product of y
.
3
There is a real number y with a property that for every real number
w,
.
Solution:
1
real number; product with every real number equals zero
2
with every real number equals zero
3
y ∗w =0
.
Speaking Mathematically
Language of Sets
Definition 1
Any collection of well-defined distinct objects or ideas is a set.
The word “well-defined” refers to a specific property which makes it easy
to identify whether the given object or idea belongs to the set or not.
The word “distinct” means that the objects of a set must be all different.
Example 2
1
The collection of children in class VII whose weight exceeds 35kg
represents a set.
2
The collection of all the intelligent children in class VII does not
represent a set because the word intelligent is vague. What may
appear intelligent to one person may not appear the same to another
person.
Speaking Mathematically
Definition 3
The objects or ideas in a set are called elements of the set.
Note: Elements are written in any order and are not repeated.
Speaking Mathematically
Notation of a Set
There is a fairly simple notation for sets. We simply list each
element (or ”member”) separated by a comma, and then put
some curly brackets around the whole thing.
Ex. {1, 2, 3, 4, 5}
The curly brackets {} are sometimes called ”set brackets” or
”braces”.
A set is usually denoted by capital letters A, B, ... and elements are
denoted by small letters a, b, ....
If x is an element of set A, then we say x ∈ A (read as either of the
following “x is an element of A”, “x belongs to A”, “x is in A”, “x
is a member of A”).
If x is not an element of set A, then we say x ∈
/ A (read as either of
the following “x is not an element of A”, “x does not belong to A”,
“x is not in A”, “x is not a member of A”).
Speaking Mathematically
Methods of Describing a Set
Roster/Listing Method/Tabular Form
In this method, a set is described by listing element, separated by
commas, within braces.
Set-Builder/Rule Method
In this method, we write down a property or rule which gives us all
the elements of the set by that rule. Moreover, all the elements of
the set must possess a single property to become a member of that
set.
In this method, the element of the set is described by using a symbol
‘x’ or any other variable followed by a colon (:) or a vertical line (|),
which is used to denote the words “such that”, then we write the
property possessed by the elements of the set, and then enclose the
whole description in braces, which stands for ‘set of all’.
Speaking Mathematically
Set-builder notation is especially useful when describing infinite sets. For
instance, in set-builder notation, the set of natural numbers greater than
7 is written as follows:
{x|x ∈ N
and x &gt; 7}
Speaking Mathematically
Example 4
Let A be the set of all vowels of English alphabets. Then, A can be
described by the following methods:
Roster/Listing Method/Tabular Form:
A = {a, e, i, o, u}
Set-Builder/Rule Method:
A = {x|x a vowel of English alphabets} or
A = {x : x a vowel of English alphabets}
In the example above, “e is an element of A (e ∈ A)”, “r is not an
element of A ( r ∈
/ A)”, and “o belongs to A (o ∈ A)”.
Speaking Mathematically
Example 5
Let B denote the set of even numbers between 6 and 14. Then, B can
be described by the following methods:
Roster/Listing Method/Tabular Form:
B = {8, 10, 12}
Set-Builder/Rule Method:
B = {x|x is an even number between 6 and 14} or
B = {x : x is an even number between 6 and 14}
Speaking Mathematically
Properties of Sets
The change in order of writing the elements does not make
any changes in the set.
In other words, the order in which the elements of a set are written
is not important. Thus, the set {a, b, c} can also be written as
{a, c, b} or {b, c, a} or {b, a, c} or {c,a,b} or {c, b, a}.
Example 6: Set A = {4, 6, 7, 8, 9} is the same as set
A = {8, 4, 9, 7, 6},i.e, {4, 6, 7, 8, 9} = {8, 4, 9, 7, 6}.
Similarly, {w , x, y , z} = {x, z, w , y } = {z, w , x, y } and so on.
Speaking Mathematically
Properties of Sets
If one or many elements of a set are repeated, the set remains
the same.
In other words, the elements of a set should be distinct. So, if any
element of a set is repeated number of times in the set, we consider
it as a single element. Thus, {1, 1, 1, 2, 2, 3, 3, 4, 4, 4} = {1, 2, 3, 4}.
Example 7: The set of letters in the word ‘GOOGLE’={G,O,L,E}.
Speaking Mathematically
Standard Notations
Notation
N
W
Z
Z+
Z−
Q
Qc
R
R+
R−
C
Description
set of all natural numbers
set of all whole numbers
set of all integers
set of all positive integers
set of all negative integers
set of all rational numbers
set of all irrational numbers
set of all real numbers
set of all positive real numbers
set of all negative real numbers
set of all complex numbers
N = {1, 2, 3, 4, ...}
W = {0, 1, 2, 3, ...}
Z = {..., −2, −1, 0, 1, 2, ...}
Z+ = {1, 2, 3, 4, ...}
−
Z = {..., −3, −2, −1}
Q = { pq : p, q ∈ Z, q 6= 0}
R = Q ∪ Qc
C = {a + ib : a, b ∈ R}
Speaking Mathematically
The set of natural numbers is also called the set of counting numbers.
The three dots &middot; &middot; &middot; are called an ellipsis and indicate that the elements of
the set continue in a manner suggested by the elements that are listed.
The integers &middot; &middot; &middot; , −3, −2, −1 are negative integers. The integers
1, 2, 3, &middot; &middot; &middot; are positive integers. Note that the natural numbers and the
positive integers are the same set of numbers. The integer zero is neither
a positive nor a negative integer.
If a number in decimal form terminates or repeats a block of digits, then
the number is a rational number. Rational numbers can also be written
in the form qp , where p and q are integers and q 6= 0.
Example 8:
1
= 0.25
4
3
= 0.27
11
are rational numbers. The bar over the 27 means that the block of digits
27 repeats without end; that is,0.27 = 0.27272727 &middot; &middot; &middot; .
Speaking Mathematically
A decimal that neither terminates nor repeats is an irrational number.
For instance, 0.35335333533335 &middot; &middot; &middot; is a nonterminating and
nonrepeating decimal and thus, is an irrational number.
Definition 9
A set containing finite number of elements or no element is called finite
set.
Example 10
The following are finite sets: {0, 1, 2}, {a, b, c, d, e}
Definition 11
A set containing infinite number of elements is called infinite set.
Example 12
The following are infinite sets: Q, N, {&middot; &middot; &middot; , −3, −2, −1, 0, 1, 2}
Speaking Mathematically
Definition 13
The number of elements in a given finite set, denoted by |A|, is called the
cardinal number of a finite set.
Example 14
If X = {2, −3, 5, 10}, then the cardinal number of X is 4, that is,
|X | = 4.
Speaking Mathematically
Definition 15
Empty set (or also called null set or void set) is a set containing no
element. It is denoted by &Oslash; or {}. The empty set is always considered as
a subset of any set.
Definition 16
A set containing a single element is called a singleton set.
Example 17
The following are singleton sets: {a}, {0}, {&Oslash;}
Speaking Mathematically
Definition 18
Let A and B be two sets. If every element of A is an element of B, then
A is called a subset of B and B is called superset of A. Written as A ⊆ B
or B ⊇ A.
Note: From the definition above, it follows that if there exists at least
one element of A which is not in B, then A is not a subset of B, written
as A * B.
Remark 19
Let A be any set. Then,&Oslash; ⊆ A and A ⊆ A. A is called an improper
subset of itself while all of its other subsets are called proper subsets. In
other words, if D is a subset of A but is not equal to A, then we say that
D is a proper subset of A, denoted by D ⊂ A.
Speaking Mathematically
Example 20
Let A = {−1, 0, 1}, B = {−2, 0, 4, 6} and C = {0, 4}. The following are
true:
C ⊆ B or B ⊇ C (since every element of C is an element of B)
A 6⊆ B (since there exists an element of A, like 1, which does not
belong to B)
&Oslash; ⊆ A,&Oslash; ⊆ C , &Oslash; ⊆ B (since empty set is always a subset of any set)
Speaking Mathematically
Example 21
Determine whether each statement is true or false.
1
{5, 10, 15, 10} ⊆ {0, 5, 10, 15, 20, 25}
2
W⊆N
3
{2, 4, 6} ⊆ {2, 4, 6}
4
&Oslash; ⊆ {x, y , z}
Solution
1
True; every element of the first set is an element of the second set.
2
False; 0 is a whole number, but 0 is not a natural number.
3
True; every set is a subset of itself.
4
True; the empty set is a subset of every set.
Speaking Mathematically
Definition 22
Two sets A and B are said to be equal, denoted by A = B, if all the
elements of set A are in set B and vice versa.
Example 23
Let A = {2, 3, 5} and B = {5, 2, 3}. Here, set A is equal to set B, that
is, A = B.
Speaking Mathematically
Definition 24
Two sets A and B are said to be equivalent, denoted by A ≡ B or
A ∼ B, if they both contain the same number of elements.
Example 25
Let A = {a, c, e} and B = {2, 4, 7}. Observe that both A and B contain
three elements. Thus, set A is equivalent to set B, that is, A ≡ B or
A ∼ B.
Remark 26
Equal sets are always equivalent.
Equivalent sets may not be equal.
Speaking Mathematically
Definition 27
A set consisting of all possible elements which occurs under consideration
is called universal set, usually denoted by U.
Example 28
Let A = {−1, 0, 1}, B = {−2, 0, 4, 6} and C = {0, 4}. Then, a universal
set of these sets can be the following sets:
U = {−2, −1, 0, 1, 4, 6} (U can be the collection of all the elements
of the considered sets)
U = {−10, −2, −1, 0, 1, 2, 4, 6} (U can also have elements that are
not in the considered sets as long as it contains all of the elements
of sets
U = R (since R contains all of the elements of sets A, B, and C )
Speaking Mathematically
Definition 29
The set formed by all the subsets of a given set A is called the power set
of A, denoted by ℘(A).
Remark 30
A power set is a set of sets. In other words, it is a collection of sets
(specifically, the subsets of the given set). Hence, the elements of a
power set are sets.
Speaking Mathematically
Remark 31
If A is a set with cardinality ‘n’, then the number of subsets of A is equal
to 2n .
Example 32
Let A = {a, b, c}. Then, the cardinality of A is 3, that is, |A| = 3. By
Remark, the number of subsets of A must be 23 = 2 &middot; 2 &middot; 2 = 8. These are
the following:
&Oslash;, A, {a}, {b}, {c}, {a, b}, {a, c}, {c, b}
Hence, the power set of A is
℘(A) = {&Oslash;, {a}, {b}, {c}, {a, b}, {a, c}, {c, b}, A}
Speaking Mathematically
Operations on Sets
1
Intersection of sets
2
Union of sets
3
Complement of the set
4
Set difference
5
Cartesian product of set
Speaking Mathematically
Intersection of Sets
Definition 33
The intersection of set A and set B is a set whose elements are those
that are common to set A and set B, denoted by A ∩ B. In set-builder
notation, we have
A ∩ B = {x : x ∈ A and x ∈ B}.
Example 34
Let A = {a, b, c, d} and B = {a, 1, 2, 4, b}. Then A ∩ B = {a, b}.
Speaking Mathematically
Definition 35
We say two sets A and B are disjoint (or mutually exclusive) if
A ∩ B = &Oslash;.
Example 36
Let A = {a, b, c, d} and B = {1, 2, 4, }. Then A ∩ B = &Oslash;. Hence, A and
B are disjoint.
Speaking Mathematically
Properties of the Intersection Operation
A∩&Oslash;=&Oslash;
Domination Law
A∩U =A
Identity Law
A∩A=A
Idempotent Law
A∩B =B ∩A
Commutative Law
A ∩ (B ∩ C ) = (A ∩ B) ∩ C
Associative Law
Speaking Mathematically
Union of Sets
Definition 37
The union of set A and set B is a set whose elements are those in A or in
B or in both, denoted by A ∪ B. If the elements are in both sets, we do
not repeat them. In set-builder notation, we have
A ∪ B = {x : x ∈ A or
x ∈ B}
Example 38
Let A = {a, b, c, d} and B = {a, 1, 2, 4, b}. Then
A ∪ B = {a, b, c, d, 1, 2, 4}.
Speaking Mathematically
Properties of the Union Operation
A∪&Oslash;=A
Identity Law
A∪U =U
Domination Law
A∪A=A
Idempotent Law
A∪B =B ∪A
Commutative Law
A ∪ (B ∪ C ) = (A ∪ B) ∪ C
Associative Law
Speaking Mathematically
The Complement of a Set
Definition 39
The complement of set A is a set whose elements are those elements of
the universal set that are not in A, denoted by Ac . In set-builder
notation, we have
Ac = {x : x ∈
/ A}.
Example 40
Let U = {1, 2, 3, 4, 5} and A = {1, 4, 5}. Hence, Ac = {2, 3}.
Speaking Mathematically
Properties of the Complement Operation
(Ac )c = A
Complement Law
A ∪ Ac = U
Complement Law
A ∩ Ac = &Oslash;
c
Complement Law
c
c
De Morgan’s Law
(A ∪ B)c = Ac ∩ B c
De Morgan’s Law
(A ∩ B) = A ∪ B
Speaking Mathematically
Set Difference Operation
Definition 41
The relative complement or set difference of sets A and B, denoted by
A − B or A \ B, is the set of all elements in A that are not in B. In
set-builder notation, we have
A − B = {x : x ∈ A and x ∈
/ B}.
Example 42
Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Then
A − B = {1, 2, 3, 4, 5} − {3, 4, 5, 6, 7} = {1, 2}
B − A = {3, 4, 5, 6, 7} − {1, 2, 3, 4, 5} = {6, 7}
Remark 43
The complement of a set A can also be written as Ac = U − A.
Speaking Mathematically
Cartesian Product Operation
Definition 44
The Cartesian product of two sets A and B, denoted by A &times; B, is the set
of all possible ordered pairs where the elements of A are first component
and the elements of B are second component. In set-builder notation,
A &times; B = {(a, b) : a ∈ A
and b ∈ B}.
Example 45
Let A = {r , t} and B = {1, 2, 3}. Then
A &times; B = {(r , 1), (r , 2), (r , 3), (t, 1), (t, 2), (t, 3)}
B &times; A = {(1, r ), (1, t), (2, r ), (2, t), (3, r ), (3, t)}.
Note that A &times; B 6= B &times; A, that is, Cartesian product is not
commutative.
Speaking Mathematically
Venn Diagrams
Venn diagrams are useful in solving simple logical problems.
Mathematician John Venn introduced the concept of representing the
sets pictorially by means of closed geometrical figures called Venn
diagrams. In Venn diagrams, the Universal Set U is represented by a
rectangle and all other sets under consideration by circles within the
rectangle. In this section, we will use Venn diagrams to illustrate various
operations like union, intersection and difference.
Speaking Mathematically
Definition 46
Venn diagrams (or set diagrams) are pictorial representations of sets
represented by closed figures.
In a Venn diagram,
a rectangle is used to represent a universal set; and
circles or ovals are used to represent other subsets of the universal
set.
Speaking Mathematically
Venn Diagrams in Different Situations
If a set B is a subset of set A, then the circle representing set B is
drawn inside the circle representing set A.
Speaking Mathematically
Venn Diagrams in Different Situations
If set A and set B have some elements in common, then to represent
them, we draw two circles which are overlapping.
Speaking Mathematically
Venn Diagrams in Different Situations
If set A and set B are disjoint, then they are represented by two
non-intersecting circles.
Speaking Mathematically
The Language of Relations and Functions
Definition 1
If A and B are two non-empty sets, then a relation R from A to B is a
subset of A &times; B . If R ⊆ A &times; B and (a, b) ∈ R, then we say that a is
related to b by the relation R, written as aRb. The set of all first
components of the ordered pairs in a relation R from a set A to a set B
is called the domain of the relation R. The set of all second components
in a relation R from a set A to a set B is called the range of the relation
R. The whole set B is called the codomain of the relation R.
Note that range is always a subset of codomain.
Speaking Mathematically
Example 2
Given R = {(x, y ) ∈ A &times; B|x = y } where A = {2, 3, 5, 7, 10} and
B = {1, 3, 5, 8, 10}. Find the domain, range, and codomain of R.
Solution:
R = {(3, 3), (5, 5), (10, 10)}
Domain={3, 5, 10}
Range={3, 5, 10}
Codomain={1, 3, 5, 8, 10}
Speaking Mathematically
Example 3
Let A = {1, 2} and B = {1, 2, 3} and define a relation R from A to B as
follows:
Given any (x, y ) ∈ A &times; B,
(x, y ) ∈ R
means that
x −y
is an integer.
2
State explicitly which ordered pairs are in A &times; B and which are in R.
Is 1R3? Is 2R3? Is 2R2?
What are the domain and co-domain of R?
Speaking Mathematically
Solution:
A &times; B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}. To determine
explicitly the composition of R, examine each ordered pair in A &times; B
to see whether its elements satisfy the defining condition in R.
(1, 1) ∈ R because
1−2
2
(1, 2) ∈
/ R because
(1, 3) ∈ R because
1−1
2
1−3
2
=
=
=
0
2
= 0, which is an integer.
−1
2 ,
−2
2
which is not an integer.
= −1, which is an integer.
(2, 1) ∈
/ R because
2−1
2
= 12 , which is not an integer.
(2, 2) ∈ R because
2−2
2
=
(2, 3) ∈
/ R because
2−3
2
=
0
2
= 0, which is an integer.
−1
2 ,
which is not an integer.
Thus
R = {(1, 1), (1, 3), (2, 2)}.
Speaking Mathematically
Yes, 1R3 because (1, 3) ∈ R.
No, 2 6 R3 because (2, 3) ∈
/ R.
Yes, 2R2 because (2, 2) ∈ R.
The domain of R is {1, 2} and the codomain is {1, 2, 3}.
Speaking Mathematically
Arrow Diagram of a Relation
Suppose R is a relation from as set A to a set B. The arrow diagram for
R is obtained as follows:
1
Represent the elements of A as points in one region and the
elements of B as points in another region.
2
For each x in A and y in B, draw an arrow from x to y if, and only
if, x is related to y by R.
Speaking Mathematically
Example 4
Let A = {1, 2, 3} and B = {1, 3, 5} and define relations S and T from A
to B as follows:
For all (x, y ) ∈ A &times; B,
(x, y ) ∈ S
means that
x &lt;y
S is a ”less than”relation.
T = {(2, 1), (2, 5)}.
Draw arrow diagrams for S and T .
Speaking Mathematically
Solution:
Speaking Mathematically
Definition 3
A function is a relation that maps each element x of a set A with one
and only one element y of another set B. In other words, it is a relation
between a set of inputs and a set of outputs in which each input is
related with a unique output.
A function f : A → B is represented as f (x) = y where (x, y ) ∈ f and
x ∈ A and y ∈ B.
```