2.1 Variables
Variable is sometimes thought of as a mathematical “John Doe” because you can use it as a placeholder when you want to talk about something; either
a) you imagine that it has one or more values but you don’t know what they are,
b) you want whatever you say about it to be equally true for all elements in a given set and you don’t want to be restricted to considering only a particular, concrete value for it.
Example of the first use of a variable:
Is there a number with the following property: doubling it and adding 3 gives the same result as squaring it?
In this statement, we introduce a variable to replace the potentially ambiguous word “it”
Is there a number x with the property that 2 x + 3 = x2 ?
The advantage of using a variable is that it allows you to give a temporary name to what you are seeking so that you can perform concrete computations with it to help discover its possible values. To emphasize the role of the variable as a placeholder, you might write the following:
Is there a number with the property that 2 × € + 3 = € 2 ?
The emptiness of the box can help you imagine filling it in with a variety of different values, some of which might make the two sides equal and others of which might not.
To illustrate the second use of variables, consider the statement:
No matter what number might be chosen, if it is greater than 2,
then its square is greater than 4.
In this case introducing a variable to give a temporary name to the (arbitrary) number you might choose enables you to maintain the generality of the statement, and replacing all instances of the word “it” by the name of the variable ensures that possible ambiguity is avoided:
No matter what number n might be chosen, if n is greater than 2, then n2 is greater than 4.
Example : Writing sentences using variables.
a. Are there numbers with the property that the sum of their squares equals the square of their sum.
b. Given any real number, its square is nonnegative.
Solution :
a) Are there numbers a and b with the property that a2 + b2 = ( a + b)2 ?
Or, Are there numbers a and b such that a2 + b2 = ( a + b)2 ?
Or, Do there exist any numbers a and b such that a2 + b2 = ( a + b)2 ?
b) Given any real number r, r2 is nonnegative.
Or, For any real number r, r2 ³ 0.
Or, For all real numbers r, r2 ³ 0.
Kinds of Mathematical Statements
1. Universal statements
2. Conditional statements
3. Existential statements
* A universal statement states that a certain property is true for all elements in a set.
Ex 1. All even numbers are multiples of 2.
Ex 2. All integers ending in 0 and 5 are divisible by 5.
* A conditional statement states that if one thing is true then some other thing also has to be true.
Ex 1. If 1296 is divisible by 12, then 1296 is divisible by 6.
Ex 2. If I am thirsty, then I will drink.
* An existential statement states that there is at least one thing for which the property is true.
Ex 1. There is an x in x2 – x – 6 = 0, such that a) x is odd, b) x is even.
Ex 2. There is a prime number that is even.
UNIVERSAL CONDITIONAL STATEMENTS
Universal statements contain some variation of the words, “For all”, and conditional statements contain versions of the words, “if-then”. A universal conditional statement is a statement that is both universal and conditional.
Ex. 1. For all animals A, if A is a dog, then A is a mammal.
A universal conditional statement can be written in ways that make them appear to be purely universal or purely conditional. The previous example can be transformed in a way that makes its conditional nature explicit but its universal nature implicit:
If A is a dog, then A is a mammal.
Or: If an animal is a dog, then the animal is a mammal.
The statement can also be expressed so as to make its universal nature explicit and its conditional nature implicit:
For all dogs A, A is a mammal.
Or : All dogs are mammals.
Rewriting a Universal Conditional Statement
Exercise 1. Fill in the blanks to rewrite the following statement.
For all real numbers x, if x is nonzero then x2 is positive.
a) If a real number is nonzero, then its square _____.
b) For all nonzero real number x, _____.
c) If x is ______, then ______.
d) The square of any nonzero real number is ______.
e) All nonzero real numbers have ______.
Answers:
a) is positive.
b) x2 is positive.
c) is a nonzero real number, x2 is positive.
d) positive.
e) positive squares( or squares that are positive).
UNIVERSAL EXISTENTIAL STATEMENTS
A 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.
Ex. Every real number has an additive inverse.
In this statement the property “has an additive inverse” applies universally to all real numbers. “Has an additive inverse” asserts the existence of something – an additive inverse – for each real numbers. The nature of the additive inverse depends on the real number, and the statement can be rewritten in several ways; some less formal and some more formal.
All real numbers have additive inverses.
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.
Introducing names for the variables simplifies references further in discussion. For instance, after the third version of the statement, you may write :
When r is positive, s is negative, when r in negative, s is positive and when r is zero, s is also zero.
One of the most important reasons for using variables in mathematics is that it gives you the ability to refer to quantities unambiguously throughout a lengthy mathematical argument, while not restricting you to consider only specific values for them.
REWRITING UNIVERSAL EXISTENTIAL STATEMENT
Fill in the blanks to rewrite the following statement:
Every pot has a lid.
a) All pots ______.
b) For all pots P, there is a ______.
c) For all pots P, there is a lid L such that _______.
Answers:
a) have lids.
b) a lid for P.
c) L is a lid for P.
EXISTENTIAL UNIVERSAL STATEMENTS
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. For example:
There is a positive integer that is less than or equal to every positive integer.
This statement is true because the number one is positive integer, and it satisfies the property of being less than or equal to every positive integer. The statement can be rewritten in several ways, some less formal and some more formal:
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.
REWRITING AN EXISTENTIAL UNIVERSAL STATEMENT
Fill in the blanks to rewrite the following statement in three different ways:
There is a person in my class who is at least as old as every person in my class.
a) Some _____ is at least as old as _____.
b) There is a person P in my class such that P is _____.
c) There is a person P in my class with the property that for every person P in my class, p is ____.
Answers:
a) person in my class, every person in my class.
b) at least as old as every person in my class.
c) at least as old as q.
Some of the most important mathematical concepts, such as the definition of a limit of a sequence, can only be defined using phrases that are universal, existential and conditional, and they require the use of all three phrases “for all”, “there is”, and “if – then”.
Example:
If a1, a2, a3, . . . is a sequence of real numbers, saying that
the limits of an as n approaches infinity is L
means that
for all positive real numbers e, there is an integer N such that
for all integers n, if n > N then –e < an – L < e .
Exercise set 2.1
Fill
in the blanks to rewrite the given statements.
1. For
all objects O, if O is a square then O has four sides.
a.
All squares _____________ .
b.
Every square ____________ .
c.
If an object is a square, then it
__________ .
d.
If O _____ , then
O _________ .
e.
For all squares O, ____________ .
2. For
all equations E, if E is quadratic then E has at most two real solutions.
a.
All quadratic equations ____________ .
b.
Every quadratic equation ___________ .
c.
If an equation is quadratic, then it _________ .
d.
If E ________ , then E _______________
.
e.
For all quadratic equation E,
__________ .
3. Every
positive number has a positive square root.
a.
All positive numbers ______ .
b.
For any positive number N, there
is _______ for N.
c.
For all positive numbers N, there
is a positive number r such that ______ .
4. Every nonzero real number has a reciprocal.
a.
All nonzero real numbers ______ .
b.
For all nonzero real numbers r, there is ______ for r.
c.
For all nonzero real numbers r, there is a real number s such that
_____.
5.
There is a real number whose product with every number leaves the number
unchanged.
a.
Some ______ has the property that its ______ .
b.
There is a real number r such that the product of r
______.
c.
There is a real number r with the property that for every real number s,
______ .
Fill in the blanks using variable or variables to rewrite the given
statements.
1. Is there an integer tha has a
remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided
by 6?
a) Is there an integer n such
that n has _______ ?
b) Does there exist ________
such that if n is divided by 5 the remainder is 2 and if ______ ?
c) Give at least one example of this
integer.
2. Given any two real numbers, there
is a real number in between.
a) Given any two real numbers x and y, there a real number r such
that r is ______ .
b) For any two ______ , _______ such that x <
r < y.
3. Is there a real number whose square
is – 1?
a) Is there a real number x such that _____ ?
b) Does there exist ______ such that x2 = – 1 ?
c) Give at least one example of this
number.
4. Given any real number, there is a
real number that is greater.
a) Given any real number r ,
there is _____ s such
that s
is ______ .
b) For any ______ , _______ such
that s > r.
5. The cube root of any negative real
number is negative.
a) Given any negative real number s , the cube root of ______ .
b) For any real number s , if
s is _____ , then _______ .
c) If a real number s
______, then ______ .
SETS
1.2 The Language of sets
Georg Cantor (1845 – 1918) introduced the word set as a formal mathematical term in 1879.
A SET is a well-defined collection of objects. The objects are the members of the elements of the set.
Examples of sets : Well-defined sets Not well-defined set
1. Set of solution to x2 – 7x + 6 = 0. 1. Set of presidents
2. Set of vowels of the English alphabet. 2. Set of subjects
3. Set of positive integers greater than 1 but less than 8 3. Set of municipalities
4. Set of cities of Region XII.
Notation:
Sets are denoted by capital letters, such as A, B, C, D, X, Y . . . whereas the lower case letters such as a, b, c, x, y . . . is used to denote members or elements of the set.
If S is a set, the notation x S, means that x is an element of S. The notation x S, means that x is not an element of S.
A set may be specified using a set-roster notation by writing all the elements between braces.
Examples :
If N is a set of countries of the UN, then the Philippines is an element of N.
If B is a set of positive even integers greater than 1 but less than 8, then 4 is an element of B.
If C is a set of cities of Region XII, then Koronadal City is an element of C.
Is Africa an element of N?
Is 5 an element of B?
Is Malaybalay City an element of C?,
A variation of the notation is used to describe a very large set, as when we write
{1, 2, 3, 4, . . . , 100}
to refer to a set of all integers from 1 to 100. A similar notation can also describe an infinite set, as when we write {1, 2, 3, . . . } to refer to a set of all integers.
The symbol . . . is called an ellipsis and is read “and so forth.”
An axiom of extension says that a set is completely determined by what its elements are – not the order in which they might be listed or the fact that some elements might be listed more than once.
Example 1 : Using the set-roster notation
a. Let A = {1, 2, 3}, B = {3, 1, 2}, and C = {1, 1, 2, 3, 3, 3}. What are the elements of A, B, and C? How are A, B, and C related ?
b. Is { 0 } = 0?
c. How many elements are in the set {1, { 1 } } ?
d. For each nonnegative integer n, let Un = { n, – n}. Find U1, U2 and U0.
Solution :
a. A, B, and C have exactly the same three elements: 1, 2, and 3. Therefore, A, B, and C are simply different ways to represent the same set.
b. {0} ¹ 0 because {0} is a set with one element, namely 0, whereas 0 is just the symbol that
represents the number zero.
c. The set {1, { 1 } } has two elements: 1 and the set whose only element is 1.
d. U1 = { 1, – 1}, U2 = { 2, – 2}, U0 = { 0, – 0} = { 0, 0} = { 0 }
SET–BUILDER NOTATION
Another way to specify a set uses a set-builder notation. Let S denote a set and let P(x) be a property that elements of S may or may not satisfy. We may define a new set to be the set of elements x in S such that P(x) is true. The set is denoted by
ORDERED PAIR
Given elements a and b, the symbol (a,
b) denotes the ordered pair consisting of a and b together with the specification
that a
is the first element of the pair and
b is the second element. Two ordered pairs (a, b) and (c, d) are equal
if, and only if a = c and b = d.
Symbolically:
(a, b) = (c, d) means that a = c and b = d.
CARTESIAN PRODUCT
Given sets A and B,
the Cartesian product of A and B, denoted by A x B
and read as “A cross B”, is the set of all ordered pairs (a, b), where a
is in A and b is in B. Symbolically:
A x B = { (a, b) | a Î A and b Î B }
Example 6: Cartesian product
Let A = { 1, 2, 3 } and B
= { m, p }
a.
Find A x B
b.
Find B x A
c.
B x B
d.
How many elements are in A x B, B x A and B x B?
e.
Let R denote the set of all real numbers.
Describe R x R.
Solution :
a.
A x B = { (1, m), (2, m), (3, m),
(1, p), (2, p), (3, p) }
b. B x A = { (m, 1), (m, 2), (m, 3), (p, 1), (p, 2), (p,
3) }
c. B x B = { (m, m), (m, p), (p, m), (p, p) }
d. A x B has six elements. This is the number of elements in A times the
number of elements in B.
B x A has six elements, the number of elements
in B times the number of elements in A.
B x B has four elements, the number
of elements in B times the number of elements in B.
e. R x R is the set of all
ordered pairs (x, y) where both x and y are real numbers. If horizontal and vertical axis are drawn on a plane
and a unit length is marked off, then each ordered pair in R x R corresponds to a unique point in
the plane, with the first and the second elements of the pair indicating the abscissa and
ordinate the point.
The term Cartesian plane is
often used to refer to a plane with this coordinate system, as shown in the figure:
1.3 The
Language of Relation and Function
There
are many kinds of relationships in the world. Two people are related by blood
if they share a common ancestor, we also have a relationship between student
and teacher, between people who work for
the same employer, and between people who have a common ethnic background.
The
objects of mathematics maybe related in many ways. A set B maybe related to a
set C if B is a subset of C, or B is not a subset of C, or B and C have a
common element. A number x is related to a number y if x < y, or if x is a factor of y,
or if x2 + y2 = 1.
Identifiers
in a computer program are related if they have the same first eight characters,
or the same memory location is used to
store their values when the program executes.
Let
A = {0, 1, 2} and B = {1, 2, 3 }. An element x in A is related to an element y
in B if, and only if, x is less than y.
The notation xRy is shorthand for
the sentence “x is related to y”. From the given set:
0 R 1
since 0 < 1
0 R 2
since 0 < 2
0 R 3
since 0 < 3
1 R 2
since 1 < 2
1 R 3 since
1 <
3 and
2 R 3
since 2 < 3
