math

1. Speaking
Mathematically
CHAPTER 2

2. Section 2.1 VARIABLES
A variable is sometimes thought of as a mathematical “John Doe” because you
can use it as a placeholder when you wan to talk about something but either:
1. you imagine that it has one or more values but you don’t know what they
are, or
2. you want whatever you say about it to be equally true for all elements is a
given set, and so you don’t want to be restricted to considering only a
particular, concrete value for it.

3. Example:
1. Is there a number with the following property: doubling it
and adding 3 gives the same result as squaring it?
2. Are there numbers with the property that the sum of their
squares equals the square of their sum?
3. Given any real number, its square is nonegative.

4. Kinds of Mathematical Statements
UNIVERSAL STATEMENT – says that a certain property is true for all elements in a set. (Ex. All
positive numbers are greater than 0)
A CONDITIONAL STATEMENT – says that if one thing is true then some other things also has to be
true. (Ex. It 378 is divisible by 18, then 378 is also divisible by 6.)
Given a property that may or may not be true, an EXISTENTIAL STATEMENT says that there is at
least one thing for which the property is true. (Ex. There is a prime number that is even.)

5. Universal statements contain some variations 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: For all animals x, if x is a dog, then x is a mammal.
If x is a dog, then x is a mammal.
Or: If an animal is a dog, then the animal is a mammal.
For all dogs x, x is mammal
Or: All dogs are mammals.
UNIVERSAL CONDITIONAL STATEMENT

6. Examples: Rewriting a Universal
Conditional Statement
Fill in the blanks to rewrite the following statement:
For all real numbers 𝑥, if 𝑥 is a nonzero then 𝑥2is positive.
a. If a real number is nonzero, then its square is ____.
b. For all nonzero real numbers 𝑥,____________.
c. If 𝑥 _______, then ________.
d. The square of any nonzero real number is ______.
e. All nonzero real numbers have _____.

7. TRY:
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 ____.

8. 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 is
existential because its second part asserts the existence of something.
Ex.
Every real number has an additive inverse.
All real numbers have additive inverses.
Or: For all real numbers r, there is an additive inverse for r.
Or: For all real number r, there is a real number s such that s is an additive
inverse for r.

9. Example: 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 ______.

10. TRY:
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 _____

11. 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 he object satisfies a certain
property for all things of a certain kind.
For ex.
There is a positive integer that is less than or equal to every positive integer.
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 or equal to m.
Or: There is a positive integer 𝑚 with the property that for all positive integers 𝑛, 𝑚 ≤ 𝑚𝑛.

12. Example: Rewriting an Existential
Universal Statement
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 q in my class, p is ____.

13. TRY:
There is a bird in this flock that is at least as heavy as every
bird in the flock.
a. Some _____ is at least as heavy as ____.
b. There is a bird b in this flock such that b is _____.
c. There is a bird b in this flock with the property that for
every bird b in the flock, b is ____.

14. Exercise 2.1 Page 29
Fill in the blanks using the variable or variables to rewrite the given statement.
1. 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?
2. Is there an integer that has a remainder 2 when it is divided by 5 and a remainder
of 3 when it is divided by 6?
a. Is 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 ____?

15. Fill in the blanks to rewrite the give statement.
8. For all objects J, if J is a square than J has four
sides.
a. All squares _____.
b. Every square ____.
c. If an object is a square, then it _____.
d. If J ___, then J ____
e. For all squares J, ____.
Exercise 2.1 Page 29
.
9. For all objects J, if J is a square than J has
four sides.
a. All squares _____.
b. Every square ____.
c. If an object is a square, then it _____.
d. If J ___, then J ____
e. For all squares J, ____.

16. 2.1 The Language of Sets
Use of the word set as a formal
mathematical term was
introduced in 1879 by George
Cantor (1845-1918)

17. A similar notation can also
describe an infinite set as when
we write, {1,2,3,…}, refers to the
set of all positive integers.
The Language of Sets
Notation: If S is a set, the notation 𝑥 ∈ 𝑆, means
than x is an element of S. The notation 𝑥 ∉ 𝑆
means x is not an element of S.
A set may be specified using the set-roster
notation by writing all of its elements between
braces.
For example, {1,2,3} denote as the set whose
elements are 1,2 and 3.
A variation of the notation is sometimes used to
describe a very large set as when we
write,{1,2,3,…,100}, refers to a set of integers from
1 to 100.

18. Examples
Using the Set-Roster Notation
1. Let 𝐴 = 1,2,3 , 𝐵 = 2,3,1 ,
𝐶 = 1 , 1, 2, 3, 3, 3, . What are the elements of
A, B and C? How are A, B, and C related?
1. 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.
2. Is {0} = 0?
2. They are not equal. Because {0} is a
set with one element, namely 0, whereas
0 is just zero.
3. How many elements are there in set {1, {1} }? 3. It has 2 elements: 1 and the set whose only
element is 1.
4. For each nonnegative integer n,
𝑙𝑒𝑡 𝑈𝑛 = 𝑛, −𝑛 . Find 𝑈1, 𝑈2, 𝑎𝑛𝑑 𝑈0.
4. 𝑈1 = 1, −1 , 𝑈2 = 2. −2 ,
𝑈0 = 0, 0 = {0}

19. Check your progress
Using the Set-Roster Notation
3. For each positive integer x, 𝑙𝑒𝑡 𝐴𝑥 = 𝑥, 𝑥2 . Find 𝐴1, 𝐴2, 𝑎𝑛𝑑 𝐴3.
2. How many elements are there in set {a, {a,b}, {a} }?
1. Let 𝑋 = 𝑎, 𝑏, 𝑐 , 𝑌 = 𝑐, 𝑎, 𝑏 , 𝑍 = 𝑎, 𝑎, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐, .
What are the elements of X, Y and Z? How are X, Y, and Z related?

20. Certain Set of Numbers that are
frequently used
Symbol Set
ℝ Set of all real numbers
ℤ Set of all integers
ℚ Set of all rational numbers, or quotients
of integers

21. 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 𝒆𝒍𝒎𝒆𝒏𝒕𝒔 𝒙 𝒊𝒏 𝑺 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 𝑷 𝒙 𝒊𝒔 𝒕𝒓𝒖𝒆.
We denote this set as follows:
{𝑥 ∈ 𝑆 𝑃 𝑥 }

22. Examples
Describe the following sets
a. {𝑥 ∈ 𝑅| − 2 < 𝑥 < 5}
b. {𝑥 ∈ 𝑍| − 2 < 𝑥 < 5}
c. {𝑥 ∈ 𝑍+| − 2 < 𝑥 < 5}
Answers
a. An open interval of real numbers
(strictly) between -2 and 5.It is
pictured using the number line.
b. {−1, 0, 1, 2, 3, 4}
c. {1, 2, 3, 4}

23. Check your progress
Describe the following sets
a. {𝑥 ∈ 𝑅| − 5 < 𝑥 < 1}
b. {𝑥 ∈ 𝑍| − 1 ≤ 𝑥 < 6}
c. {𝑥 ∈ 𝑍−| − 4 ≤ 𝑥 ≤ 0}

24. Subsets
If A and B are sets, then A is called a subset of B, written 𝐴 ⊆ 𝐵, if
and only if, every element of A is also an element of B.
𝐴 ⊆ 𝐵 means that For all elements x, if 𝑥 ∈ 𝐴, 𝑡ℎ𝑒𝑛 𝑥 ∈ 𝐵
𝐴 ⊆ 𝐵 means that For all elements x, if 𝑥 ∈ 𝐴, 𝑡ℎ𝑒𝑛 𝑥 ∉ 𝐵
𝐴 𝑖𝑠 𝑎 𝒏𝒐𝒕 𝒂 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐵 means that there is at least one element of A
that is not an element of B.
𝐴 𝑖𝑠 𝒑𝒓𝒐𝒑𝒆𝒓 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐵 if, and only if, every element of B is in B but
there is at least one element of B that is not in A.

25. Examples
Let 𝑨 = 𝒁+,
𝑩 = 𝒏 ∈ 𝒁 0 ≤ 𝑛 ≤ 100 , 𝑎𝑛𝑑
C = 100,200,300,400,500 .
Evaluate the truth and falsity of the
following statements:
a. 𝐵 ⊆ 𝐴
b. C is a proper subset of A
c. C and B have at least one element in
common
d. 𝐶 ⊆ 𝐵
e. 𝐶 ⊆ 𝐶
Answers
a. False. Zero is not a positive integer.
Thus 0 is in B, but it is not in A. 𝐵 ⊆ 𝐴
b. 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.
c. True . For example, 100 is in both C
and B
d. False . For example, 200 is in C but not
in B
e. True . Every element in C is in C.

26. Check your progress
a. A ⊆ 𝐵
b. 𝐵 ⊆ 𝐴
c. A is a proper subset of B
d. 𝐶 ⊆ 𝐵
e. C is a proper subset of A
Let 𝑨 = {𝟐, 𝟐 , 𝟐)𝟐 , 𝑩 = {2, 2 , 2 }, 𝑎𝑛𝑑 𝐶 = {2}
Evaluate the truth and falsity of the following statements:

27. Examples
Distinction between ∈ , ⊆
a. 2 ∈ {1,2,3}
b. 2 ∈ 1, 2,3
c. 2 ⊆ 1, 2, 3
d. 2 ⊆ 1, 2, 3
e. 2 ⊆ 1 , 2
f. 2 ∈ { 1 , 2 }

28. Cartesian Product
Given sets A and B, the Cartesian product of A and B, denoted AxB
read “A cross B”, is the set of all ordered pairs (a, b), where a is in A
and b is in B.
Symbolically: AxB ={ (a,b)/ a  A and b  B }

29. Examples
Cartesian Products
Let 𝐴 = 1, 2, 3 , 𝑎𝑛𝑑 𝐵 = 𝑢, 𝑣 , 𝑓𝑖𝑛𝑑
a. AxB
b. BxA
c. BxB
d. How many elements are in AxB, BxA,
and BxB?
Answers

30. Check your progress
Cartesian Products:
Let Y = 𝑎, 𝑏, 𝑐 , 𝑎𝑛𝑑 𝑍 = 1,2 , 𝑓𝑖𝑛𝑑
a. Y x Z
b. Z x Y
c. Y x Y
d. How many elements are in Y x Z, Z x Y, and Y
x Y?

31. Language of Relations
and Functions
SECTION2.3
PAGE 39 (MATHEMATICS IN THE MODERN WORLD)
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Eg1qY91bc4
https://www.youtube.com/watch?v=U
z0MtFlLD-k

32. 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
xRy, if , and only if, (x,y) is in R. The set A is called the domain of R
and the set B is called the co-domain.
The notation for relation R may be written as follows:
xRy means that (x,y)  R.
The notation xRy means that x is not related to y by R.
xRy means that (x,y)  R.

33. Example:
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. Find the elements of R.
b. What are the domain and co-domain of R?
c. Construct the arrow diagram of R.

34. Solutions
List the elements of A x B = { (1,1) , (1,2), (1,3), (2,1), (2,2), (2,3) }
Examine each ordered pair in AxB to see whether its elements satisfy
the defining condition for R.
(1,1) ; x+y2 =1+12 = 22 =1, 1 is an integer, (1,1) R or 1R1.
𝑎. 𝑅 = { 1,1 , 1,3 , 2,2 }

35. Answers:
𝑎. 𝑅 = { 1,1 , 1,3 , 2,2 }
𝑏. 𝐷𝑜𝑚𝑎𝑖𝑛 = {1, 2}
𝑏. 𝐶𝑜 − 𝐷𝑜𝑚𝑎𝑖𝑛 = {1, 2,3}
𝑐. 𝐴𝑟𝑟𝑜𝑤 𝐷𝑖𝑎𝑔𝑟𝑎𝑚

36. Check your Progress
Let 𝑌 = 0,1,2 𝑎𝑛𝑑 𝑍 = {0,1} and define a relation R from Y to Z as follows:
Given any 𝑥, 𝑦 ∈ 𝑌 × 𝑍,
𝑥, 𝑦 ∈ 𝑅 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡
𝑥 + 𝑦
2
𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟.
a. Find the elements of R
b. What are the domain and co-domain of R?
c. Construct an arrow diagram of R.

37. Functions
A function F from set A to 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.

38. **A relation F from A to B is a function if, and only if:
1. Every element of A is the first element of an ordered pair of F.
2. No two distinct ordered pairs in F have the same first element.
Notation: If A and B are sets and F is a function from A to
B, then given any element x in A, the unique element in B is
related to x by F is denoted F(x), which is read “F of x.”
Functions

39. Example:
1. Given relation R = { (1,2) , (2,2) , (3, 3 ) }
2. Given relation T = { ( 1, 2) , ( 1,3) , (2, 4) }
T is not a function(one to many)
R is a function (many to one)
3. Let 𝐴 = 2,4,6 𝑎𝑛𝑑 𝐵 = {1,3,5}

40. Example:
3. Let 𝐴 = 2,4,6 𝑎𝑛𝑑 𝐵 = 1,3,5 . Which of the relations R, S, and
T defined below are functions from A to B?
a. 𝑅 = 2,5 , 4,1 , 4,3 , 6,5
b. For all 𝑥, 𝑦 ∈ 𝐴 × 𝐵, 𝑥, 𝑦 ∈ 𝑆 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑦 = 𝑥 + 1
c. T defined by the arrow diagram
2
4
6
1
3
5

41. Answers:
1. R is not a function because it does not satisfy property (2). The
ordered pairs (4,1) and (4,3) have the same first element.
2. S is not a function because it does not satisfy property (1).
3. T is a function.

42. Check your Progress
Let 𝑋 = 𝑎, 𝑏, 𝑐 𝑎𝑛𝑑 𝑦 = {1,2,3,4} . Which of the relations A, B, C
defined below are functions from X to Y
a. 𝐴 = 𝑎, 1 , 𝑏, 2 , 𝑐, 3
b. For all 𝑥, 𝑦 ∈ 𝑋 × 𝑌, 𝑥, 𝑦 ∈ 𝐵 𝑚𝑒𝑎𝑛𝑠 𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝑒𝑣𝑒𝑛
c. C is defined by the arrow diagram
a
b
c
1
2
3
4

43. End!
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Eg1qY91bc4
https://www.youtube.com/watch?v=U
z0MtFlLD-k