This document introduces some key concepts in set theory and mathematical language. It defines what a set is, and introduces ways to represent sets using roster notation. It discusses well-defined sets and the symbols to denote whether an element is or is not part of a set. Examples are provided to illustrate definitions of natural numbers, integers, rational numbers, and real numbers. The document also covers universal and existential statements, and how to rewrite them using variables.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Roots and Coefficients of Quadratic Equations. It also discusses and explains the rules, steps and examples of Roots and Coefficients of Quadratic Equations
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We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq 0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq W \subseteq Z \subseteq Q.
sets of numbers and interval notation, operation on real numbers, simplifying expression, linear equation in one variable, aplication of linear equation in one variable, linear equation and aplication to geometry, linear inequaloties in one variable, properties of integers exponents and scientific notation
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
2. Speaking Mathematically
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.
3. Speaking Mathematically
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.
4. 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?
5. 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.
6. 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?
7. 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.
8. 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.
9. 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.
10. Universal Conditional Statements
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
11. Universal Conditional Statements
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.
12. 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 _______.
13. 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 is positive.
14. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
b. For all nonzero real numbers x, ___________.
15. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
b. For all nonzero real numbers x, x2 is positive.
16. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
c. If x ____________________ then _________.
17. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
c. If x is a nonzero real number, then x2 is positive.
18. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
d. The square of any nonzero real number is________.
19. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
d. The square of any nonzero real number is positive.
20. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
e. All nonzero real numbers have _________.
21. Rewriting a Universal Conditional Statement
For all real numbers x, if x is nonzero then x is positive.
e. All nonzero real numbers have positive squares (or: squares
that are positive).
22. 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 __________.
23. 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.
25. 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.
26. Universal Existential Statements
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.
27. Universal Existential Statements
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.
28. 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 ______.
29. 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.
30. 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 _______.
33. 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.
36. 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.
37. 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
38. 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.
40. 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.
43. Use The Roster Method to Represent a Set of Numbers
a. The set of natural numbers less than 5
44. 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.
45. 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. Using the roster method, we
write this set as {1, 2, 3, 4}.
46. Use The Roster Method to Represent a Set of Numbers
b. The solution set of x + 5 = -1
47. Use The Roster Method to Represent a Set of Numbers
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}.
48. Use The Roster Method to Represent a Set of Numbers
c. The set of negative integers greater than -4
49. Use The Roster Method to Represent a Set of Numbers
c. The set of negative integers greater than -4
The set of negative integers greater than -4 is {-3, -2, -1}.
50. Use The Roster Method to Represent a Set of Numbers
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
51. Use The Roster Method to Represent a Set of Numbers
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)
52. Definitions Regarding Sets
A set is well defined if it is possible to determine whether any
given item is an element of the set.
53. 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.
54. 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.
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.
55. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as
56. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as 4 ∈ N.
57. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as 4 ∈ N. The symbol ∈ is
read
58. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as 4 ∈ N. The symbol ∈ is
read “is an element of.”
59. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as 4 ∈ N. The symbol ∈ is
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,
60. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as 4 ∈ N. The symbol ∈ is
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, ∈, and
write
61. Definitions Regarding Sets
The statement “4 is an element of the set of natural numbers”
can be written using mathematical notation as 4 ∈ N. The symbol ∈ is
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, ∈, and
write -3 ∈ N.
63. Apply Definitions Regarding Sets
Determine whether each statement is true or false.
a. 4 ∈ {2, 3, 4, 7}
Since 4 is an element of the given set, the statement is true.
65. Apply Definitions Regarding Sets
Determine whether each statement is true or false.
b. -5 ∈ N
There are no negative natural numbers, so the statement is false.
67. Apply Definitions Regarding Sets
Determine whether each statement is true or false.
c.
1
2
∈ I
Since
1
2
is not an integer, the statement is true.
68. Apply Definitions Regarding Sets
Determine whether each statement is true or false.
d. The set of nice cars is a well-defined set.
69. Apply Definitions Regarding Sets
Determine whether each statement is true or false.
d. The set of nice cars is a well-defined set.
The word nice is not precise, so the statement is false
70. Apply Definitions Regarding Sets
Determine whether each statement is true or 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
71. Apply Definitions Regarding Sets
Determine whether each statement is true or false.
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
75. Apply Definitions Regarding Sets
Set-builder notation is especially useful when describing infinite sets.
76. Apply Definitions Regarding Sets
In set-builder notation, the set of natural numbers greater than 7 is
written as follows:
77. Apply Definitions Regarding Sets
In set-builder notation, the set of natural numbers greater than 7 is
written as follows:
{x | x ∈ N and x > 7}
78. Apply Definitions Regarding Sets
In set-builder notation, the set of natural numbers greater than 7 is
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
79. Apply Definitions Regarding Sets
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
80. 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
81. 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
{x|x ∈ I and x ˃ -3}.
82. Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
b. The set of whole numbers less than 1000
83. Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
b. The set of whole numbers less than 1000
{x|x ∈ W and x ˂ 1000}
84. Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
c. The set of integers less than 9
85. Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
c. The set of integers less than 9
{x|x ∈ I and x ˂ 9}.
86. Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
d. The set of natural numbers greater than 4
87. Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
d. The set of natural numbers greater than 4
{x|x ∈ N and x ˃ 4}
89. Apply Definitions Regarding Sets
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.
90. Apply Definitions Regarding Sets
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.
91. Apply Definitions Regarding Sets
In this case, A has a cardinal number of 4, which is sometimes
stated as “A has a cardinality of 4.”
92. The Cardinality of a Set
Find the cardinality of each of the following sets.
a. J = {2, 5}.
93. The Cardinality of a Set
Find the cardinality of each of the following sets.
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.
94. The Cardinality of a Set
Find the cardinality of each of the following sets.
b. S = {3, 4, 5, 6, 7, ..., 31}
95. The Cardinality of a Set
Find the cardinality of each of the following sets.
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.
96. The Cardinality of a Set
Find the cardinality of each of the following sets.
c. T = {3, 3, 7, 51}
97. The Cardinality of a Set
Find the cardinality of each of the following sets.
c. T = {3, 3, 7, 51}
Elements that are listed more than once are counted only once.
Thus n(T) = 3.
98. The Cardinality of a Set
Find the cardinality of each of the following sets.
d. C = {-1, 5, 4, 11, 13}
99. The Cardinality of a Set
Find the cardinality of each of the following sets.
d. C = {-1, 5, 4, 11, 13}
n(C) = 5
100. The Cardinality of a Set
Find the cardinality of each of the following sets.
e. D = {0}
101. The Cardinality of a Set
Find the cardinality of each of the following sets.
e. D = {0}
n(D) = 1
102. The Cardinality of a Set
Find the cardinality of each of the following sets.
f. E = ꓳ
103. The Cardinality of a Set
Find the cardinality of each of the following sets.
f. E = ꓳ
n(E) = 0
104. Apply Definitions Regarding Sets
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.
105. Apply Definitions Regarding Sets
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.
106. 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}
107. 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}
The sets are not equal. However, each set has exactly five
elements, so the sets are equivalent.
108. Equal Sets and Equivalent Sets
State whether each of the following pairs of sets are equal, equivalent,
both, or neither.
b. {4, -2, 7}, {3, 4, 7, 9}
109. Equal Sets and Equivalent Sets
State whether each of the following pairs of sets are equal, equivalent,
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
110. 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.
111. 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
112. 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
False, Zero is not a positive integer. Thus, zero is in B but zero is not in a
and so B A.
113. 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.
b. C is a proper subset of A
114. 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.
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.
115. 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.
c. C and B have at least one element in common
116. 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.
c. C and B have at least one element in common
True, for example, 100 is both B and C
117. 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.
d. C B
118. 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.
d. C B
False, for example, 200 is in C but not in B.
119. 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.
e. C C
120. Apply the Definition of Subsets
Determine whether each statement is true or false.
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}
121. Apply the Definition of Subsets
Solution
a. (5, 10, 15, 20} {0, 5, 10, 15, 20, 25, 30}
125. 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.
126. 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
127. 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
128. 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)?
129. 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 ?
130. 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
131. The Language of Relations
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
132. The Language of Relations
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)}
133. The Language of Relations
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.
134. 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?
135. 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.
136. 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.
137. 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.