Chapter 2 - Speaking Mathematically.pdf

Speaking Mathematically
Chapter 2

The aim is to introduce to you a mathematical way of thinking that can
serve you in a wide variety of situations. Often when you start work on
a mathematical problem, you may have only a vague sense on how to
proceed. You may begin at looking examples, drawing pictures, playing
around with notations, rereading the problem to focus on more in its
details, and so forth. The closer you get a solution, the more your
thinking has to crystalize. And the more you need to understand, the
more you need language that expresses mathematical ideas clearly,
precisely and unambiguously.

This will introduce to you to some of the special language that is
a foundation for much mathematical thought, the language of
variables, sets, relations and functions.

Variables
A variable is used as a placeholder when you want to talk about
something:
1. it has one or more values you don’t know what they are;
Ex. Is there a number with the following property: doubling it and
adding 3 gives the same result, as squaring it?

Variables
A variable is used as a placeholder when you want to talk about
something:
2. you want whatever you say about it to be equally true for all
elements in a given set
Ex. No matter what number might be chosen, if it is greater than 2 then
n2 is greater than four.

Writing Sentences Using Variables
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?
Are there numbers a and b with that 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?

Writing Sentences Using Variables
Use variables to rewrite the following sentences more formally.
2. Given any real number, its square is nonnegative.
Given any real number r, 𝑟2 ≥ 0.
Or. For any real number r, 𝑟2 ≥ 0.
Or. For all real number r, 𝑟2 ≥ 0.

Some important kinds of Mathematical Statements
1. Universal statement a certain property is true for all elements in a
set.
Ex. All positive numbers are greater than zero.
2. Conditional statement one thing is true then the other thing has also
to be true.
Ex. If 378 is divisible by 18, then 378 is divisible by 6.

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.
For all animals a, if a is a dog, then a is a mammal.

One of the most important facts about universal conditional
statements is that they be rewritten in ways that make them appear to
be purely universal or purely conditional. The previous statement can
be rewritten 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.
All dogs are mammals.

Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
a. If a real number is nonzero, then its square _______.

a. If a real number is nonzero, then its square is positive.

b. For all nonzero real numbers x, ___________.

b. For all nonzero real numbers x, x2 is positive.

c. If x ____________________ then _________.

c. If x is a nonzero real number, then x2 is positive.

d. The square of any nonzero real number is________.

d. The square of any nonzero real number is positive.

e. All nonzero real numbers have _________.

e. All nonzero real numbers have positive squares (or: squares
that are positive).

Check Your Progress
For all real numbers x, if x is greater than 2, then x2 is greater than 4.
a. If a real number is greater than 2, then its square is __________.
b. For all real numbers greater than 2, ____________.
c. If x ______, then ______.
d. The square of any real number greater than 2 is__________.
e. All real numbers greater than 2 have __________.

Some important kinds of Mathematical Statements
Existential statement there is at least one thing for which the property
is true.
Ex. There is a prime number which is even.
A prime number is a natural number greater than 1 that cannot be
formed by multiplying two smaller natural number.

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.
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 number.
However, the nature of the additive inverse depends on the real
number; different real numbers have different additive inverses.
Knowing that an additive inverse is a real number, you can rewrite this
statement 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.

Introducing names for the variables simplifies references in further
discussion. For instance, after the third version of the statement you
might go on to write: When r is positive, s is negative, when r is
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 a Universal Existential Statement
Every pot has a lid.
a. All pots _______.
b. For all pots P, there is _____.
c. For all pots P, there is a lid L such that ______.

Rewriting a Universal Existential Statement
Every pot has a lid.
a. All pots have lids.
b. For all pots P, there is a lid for P.
c. For all pots P, there is a lid L such that L is a lid for P.

CHECK YOUR PROGRESS
All bottles have cap.
a. Every bottle______ .
b. For all bottles B, there________.
c. For all bottles B, there is a cap C such that _______.

Basic Number Sets
Natural Numbers or Counting Numbers N = {1, 2, 3, 4, 5, …}

Basic Number Sets
The set of natural numbers is also called the set of counting
numbers. The three dots ... are called an ellipsis and indicate that the
elements of the set continue in a manner suggested by the elements
that are listed.

Basic Number Sets
Whole Numbers W = {0,1, 2, 3, 4, 5, …}

Basic Number Sets
Integers I = { …, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Basic Number Sets
The integers ..., -4, -3, -2, -1 are negative integers. The integers 1,
2, 3, 4, ... 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.

Basic Number Sets
Rational Numbers Q = the set of all terminating or repeating numbers
Irrational Numbers I = the set of all nonterminating, nonrepeating
decimals
Real Numbers R = the set of all rational and irrational numbers

Basic Number Sets
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
𝑝
𝑞
, where p and q are integers and q ≠ 0.

Basic Number Sets
Example,
1
4
= 0.25 and
3
11
= 0.27 are rational numbers.

Basic Number Sets
Example,
1
4
= 0.25 and
3
11
= 0.27 are rational numbers.
The bar over the 27 means that the block of digits 27 repeats without
end; that is, 0.27 = 0.27272727.... A decimal that neither terminates
nor repeats is an irrational number. For instance, 0.35335333533335...
is a nonterminating, nonrepeating decimal and thus is an irrational
number.

Basic Number Sets
Real Numbers R = the set of all rational and irrational numbers

Basic Number Sets
Every real number is either a rational number or an irrational number.

Use The Roster Method to Represent a Set of Numbers
a. The set of natural numbers less than 5

The set of natural numbers is given by {1, 2, 3, 4, 5, 6, 7, ...}. The natural
numbers less than 5 are 1, 2, 3, and 4.

The set of natural numbers is given by {1, 2, 3, 4, 5, 6, 7, ...}. The natural
numbers less than 5 are 1, 2, 3, and 4. Using the roster method, we
write this set as {1, 2, 3, 4}.

b. The solution set of x + 5 = -1

b. The solution set of x + 5 = -1
Adding -5 to each side of the equation produces x = -6. The solution set
of x + 5 = -1 is {-6}.

c. The set of negative integers greater than -4

c. The set of negative integers greater than -4
The set of negative integers greater than -4 is {-3, -2, -1}.

Use the roster method to write each of the given sets.
a. The set of whole numbers less than 4
b. The set of counting numbers larger than 11 and less than or equal
to 19
c. The set of negative integers between -5 and 7

Use the roster method to write each of the given sets.
a. The set of whole numbers less than 4
(0, 1, 2, 3)
b. The set of counting numbers larger than 11 and less than or equal to
19
(12,13,14, 15, 16, 17, 18, 19)
c. The set of negative integers between -5 and 7
(-4, -3, -2, -1)

Deﬁnitions Regarding Sets
A set is well defined if it is possible to determine whether any
given item is an element of the set.

For instance, the set of letters of the English alphabet is well
defined.

For instance, the set of letters of the English alphabet is well
defined.
The set of great songs is not a well-defined set. It is not possible
to determine whether any given song is an element of the set or is not
an element of the set because there is no standard method for making
such a judgment.

The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as

can be written using mathematical notation as 4 ∈ N.

can be written using mathematical notation as 4 ∈ N. The symbol ∈ is
read

read “is an element of.”

read “is an element of.” To state that “-3 is not an element of the set of
natural numbers,” we use the “is not an element of” symbol,

natural numbers,” we use the “is not an element of” symbol, ∈, and
write

natural numbers,” we use the “is not an element of” symbol, ∈, and
write -3 ∈ N.

Apply Deﬁnitions Regarding Sets
Determine whether each statement is true or false.
a. 4 ∈ {2, 3, 4, 7}

a. 4 ∈ {2, 3, 4, 7}
Since 4 is an element of the given set, the statement is true.

b. -5 ∈ N

b. -5 ∈ N
There are no negative natural numbers, so the statement is false.

c.
1
2
∈ I

c.
1
2
∈ I
Since
1
2
is not an integer, the statement is true.

d. The set of nice cars is a well-defined set.

d. The set of nice cars is a well-defined set.
The word nice is not precise, so the statement is false

a. 5.2 ∈ {1, 2, 3, 4, 5, 6}
b. -101 ∈ I
c. 2.5 ∈ W
d. The set of all integers larger than ꙥ is a well-defined set

a. 5.2 ∈ {1, 2, 3, 4, 5, 6} false
b. -101 ∈ I true
c. 2.5 ∈ W true
d. The set of all integers larger than ꙥ is a well-defined set. true

The empty set, or null set, is the set that contains no elements.

The symbol ꓳ or { } is used to represent the empty set.

Another method of representing a set is set-builder notation.

Set-builder notation is especially useful when describing infinite sets.

In set-builder notation, the set of natural numbers greater than 7 is
written as follows:

written as follows:
{x | x ∈ N and x > 7}

written as follows:
{x | x ∈ N and x > 7}
the of all such x is an element and x is
set elements x that of the set of greater than 7
natural numbers

membership conditions
{x | x ∈ N and x > 7}
the of all such x is an element and x is
set elements x that of the set of greater than 7
natural numbers
The preceding set-builder notation is read as “the set of all elements x
such that x is an element of the set of natural numbers and x is greater
than 7.” It is impossible to list all the elements of the set, but set-
builder notation defines the set by describing its elements

Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
a. The set of integers greater than -3

a. The set of integers greater than -3
{x|x ∈ I and x ˃ -3}.

b. The set of whole numbers less than 1000

b. The set of whole numbers less than 1000
{x|x ∈ W and x ˂ 1000}

c. The set of integers less than 9

c. The set of integers less than 9
{x|x ∈ I and x ˂ 9}.

d. The set of natural numbers greater than 4

d. The set of natural numbers greater than 4
{x|x ∈ N and x ˃ 4}

A set is finite if the number of elements in the set is a whole
number.

A set is finite if the number of elements in the set is a whole
number. The cardinal number of a finite set is the number of elements
in the set.

The cardinal number of a finite set A is denoted by the notation
n(A). For instance, if A = {1, 4, 6, 9}, then n(A) = 4.

In this case, A has a cardinal number of 4, which is sometimes
stated as “A has a cardinality of 4.”

The Cardinality of a Set
Find the cardinality of each of the following sets.
a. J = {2, 5}.

a. J = {2, 5}.
Set J contains exactly two elements, so J has a cardinality of 2.
Using mathematical notation, we state this as n(J) = 2.

b. S = {3, 4, 5, 6, 7, ..., 31}

b. S = {3, 4, 5, 6, 7, ..., 31}
Only a few elements are actually listed. The number of natural
numbers from 1 to 31 is 31. If we omit the numbers 1 and 2, then the
number of natural numbers from 3 to 31 must be 31 - 2 = 29.
Thus n(S) = 29.

c. T = {3, 3, 7, 51}

c. T = {3, 3, 7, 51}
Elements that are listed more than once are counted only once.
Thus n(T) = 3.

d. C = {-1, 5, 4, 11, 13}

d. C = {-1, 5, 4, 11, 13}
n(C) = 5

e. D = {0}

e. D = {0}
n(D) = 1

f. E = ꓳ

f. E = ꓳ
n(E) = 0

Equal Sets
Set A is equal to set B, denoted by A = B, if and only if A and B have
exactly the same elements.

Equivalent Sets
Set A is equivalent to set B, denoted by A~ B, if and only if A and B have
the same number of elements.

Equal Sets and Equivalent Sets
State whether each of the following pairs of sets are equal, equivalent,
both, or neither.
a. {a, e, i, o, u}, {3, 7, 11, 15, 19}

both, or neither.
a. {a, e, i, o, u}, {3, 7, 11, 15, 19}
The sets are not equal. However, each set has exactly five
elements, so the sets are equivalent.

both, or neither.
b. {4, -2, 7}, {3, 4, 7, 9}

both, or neither.
b. {4, -2, 7}, {3, 4, 7, 9}
The first set has three elements and the second set has four
elements, so the sets are not equal and are not equivalent

Subsets
Subsets () a set of all the elements are contained in another set.
If A and B are sets, Then A is called subset of B, A  B, if and only if
every element of A is also element of B.
Proper Subset of a set is a subset of that is not equal to. In other words,
if a proper subset of, then all elements of are in but contains at least
one element that not in.
For example, if A = {1, 3, 5} then B = {1, 5} then B is a proper subset of
A.

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
Evaluate the truth or falsity of each of the following statements.
a. B  A

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
a. B  A
False, Zero is not a positive integer. Thus, zero is in B but zero is not in a
and so B  A.

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
b. C is a proper subset of A

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
b. C is a proper subset of A
True, Each element in C is a positive integer and hence, is in A, but
there are elements in A that are not in C.

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
c. C and B have at least one element in common

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
c. C and B have at least one element in common
True, for example, 100 is both B and C

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
d. C  B

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
d. C  B
False, for example, 200 is in C but not in B.

Subsets
Let A = Z+, B = {n ∈ Z ꟾ0≤n≤100}, and C ={100, 200. 300, 400, 500}.
e. C  C

Apply the Definition of Subsets
a. (5, 10, 15, 20}  {0, 5, 10, 15, 20, 25, 30}
b. W  N
c. {2, 4, 6}  (2, 4, 6}
d. ꓳ  {1, 2, 3}

Solution
a. (5, 10, 15, 20}  {0, 5, 10, 15, 20, 25, 30}

Solution
b. W  N

Solution
c. {2, 4, 6}  (2, 4, 6}

Solution
d. ꓳ  {1, 2, 3}

Proper Subsets of a Set
Proper Subset
Set A is a proper subset of set B, denoted by A  B, if every
element of A is an element of B, but there is at least one element in B
that is not in A and A ≠ B.
To illustrate the difference between subsets and proper subsets,
consider the following two examples.
1. Let R = {Mars, Venus} and S = {Mars, Venus, Mercury}. The first set, R,
is a subset of the second set, S, because every element of R is an
element of S. In addition, R is also a proper subset of S, because R ≠ S.
2. Let T = {Europe, Africa} and V = {Africa, Europe}. The first set, T, is a
subset of the second set, V; however, T is not a proper subset of V
because T = V.

Proper Subsets of a Set
For each of the following, determine whether the first set is a proper
subset of the second set.
a. {a, e, i, o, u}, {e, i, o, u, a}
b. N, I
Solution
a. Because the sets are equal, the first set is not a proper subset of the
second set.
b. Every natural number is an integer, so the set of natural numbers is
a subset of the set of integers. The set of integers contains elements
that are not natural numbers, such as -3. Thus the set of natural
numbers is a proper subset of the set of integers

Ordered Pair
Ordered pair are number written in a certain order. Usually written in
parentheses ( ).
Given element a and b , the symbol (a, b) denotes the order pair
consisting a and b together with the specification that a is the first
element and b is the second element.
a. Is (1, 2) = (2, 1)?
No, By definition of equality of ordered pair.
(1, 2) = (2, 1) if, and only if, 1 =2 and 2 = 1.
b. Is (3,
5
10
) = ( 9,
1
2
)?
Yes, By definition of equality of ordered pair. 3 = 9 ,
5
10
=
1
2

Ordered Pair
c. What is the first element of (1, 1)?
In the ordered pair (1, 1), the first and second element is both 1.
d. Is (0, 10) = (10, 0)?
e. Is (4, 33) = (22,27)?
f. What is the first element of (2, 5)?

Cartesian product
Cartesian product is the product of two sets: the product of set X and
set Y is the sets contains all ordered pairs (x, y) for which x belongs to X
and y belongs to Y.
Let A = (1, 2, 3) and B = (u, v)
a. Find A x B.
b. Find B x A.
c. Find B x B.
d. How many elements in A x B, B x A, and B x B ?

The Language of Relations
Let A = {0, 1, 2} and B = {1, 2 3} and let us say that an element x
in A is related to an element y in B if, and only if, x is less than y. Let us
use the notation x R y as a shorthand for a sentence “x is related to y.”
Then
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

On the other hand, if the notation x R y represents the sentence “x is
not related to y,” then
1 R 1 since 1 ˂ 1
2 R 1 since 2 ˂ 1
2 R 2 since 2 ˂ 2

Recall the cartesian product of A and B, AxB, consist of all ordered pairs
whose first element is in A and whose second element is in B.
AxB = {(x,y) |x ∈ A and y ∈ B}
In this case,
AxB = {(0, 1), (0, 2), (0,3), (1, 1), (1, 2), (1, 3), (2,1), (2, 2),(2,3)}.
The elements of some ordered pairs in AxB are related, whereas the
elements of other ordered pairs are not. All oredered pairs in AxB
whose elements are related
{(0, 1), (0, 2), (0,3), (1, 2), (1, 3),(2,3)}

Relation
Let A and B be sets. A relation R from A to B is a subset of AxB. Given an
ordered pair (x, y) in AxB, x is related to y by R, written x R y, if, and only
if, (x, y) is in R. The set A is called the domain of R and the set B is called
its co-domain.
The notation for a relation R may be written symbolically as follows:
x R y means that (x, y) ∈ R.
The notation x R y means that x is not related to y by R:
x R y means that (x, y) ∈ R.

Example 1. A Relation as a Subset
Let A ={1,2} and B={1,2, 3} and define a relation R from A to B as
follows:
Given any (x, y) ∈ AxB,
(x, y) ∈ R means that
𝑥−𝑦
2
is an integer.
a. State explicitly which ordered pairs are in A x B and which are in R.
b. Is 1 R 3? Is 2 R 3? Is 2 R 2?
c. What are the domain and co-domain of R?

Chapter 2 Summary
2.1 Variables
Universal Conditional Statement 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.
Universal Existential Statement 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.
Existential Universal Statement 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.

Chapter 2 Summary
2.2 The Language of Sets
Set-Roster Notation. A set may be specified using the set-roster notation by writing
all of its elements between braces,
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 all elements x in
S such that P(x) is true. We denote this set as follows: x ∈ SIP(x)}
Subset. If A and B are sets, then A is called a subset of B, written ACB, if, and only if,
every element of A is also an element of B.
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.
Cartesian Product. Given sets A and B, the Cartesian product of A and B, denoted A
x B and read A cross B," is the set of all ordered pairs (a, b), where a is in A and b is
in B.

Chapter 2 Summary
2.3 The Language of Relations and Functions
Relation Let A and B be sets. A relation R from A to B 1S a subset of A x
B. Given an ordered pair (x, y) in A x B, x is related toy by R, written x R
y, if, and only if, (x, y) is in R. The set A is called the domain of R and the
set B is called its co-domain.
Function. A function F from a set A to a set B is a relation with domain
A and co-domain B that satisfies the following two properties:
1. For every element x in A, there is an element y in B such that (x, y) ∈
F
2. For all elements x in A and y and z in B, if (x, y) ∈ F and (x, z) ∈ F, then
y =z.

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