lecture 16 quantitative reasoning.pptx

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Quantitative Reasoning
By
Nazia Aslam
Designation: Lecturer
Niazi Medical & Dental College
Sargodha

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Objectives
• Sets and their Operations
• Definition
• Ways of expressing sets
• Types of sets
• The size of the set

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• Introduction: the fundamental discrete
structure on which all other discrete structure
are built, namely the sets. Sets are used to
group objects together. But not always the
objects in a set have similar properties. All the
students who are currently enrolled in your
school make up a set.
Sets and their Operations

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Definition
• A set is defined as the collection of well
defined distinct elements. A set is usually
denoted by capital letters say A, B, C ….. etc.,
and the elements of the set are denoted by
small letter sat a, b, c , …. etc., the elements of
the sets are enclosed by braces {}. The
following are the example of sets. A= {x,y,z} B=
{1,2,3,4,5}.

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• Sets: a set is an unordered collection of
objects called elements or numbers of the
sets. A set is said to contain its elements. We
write a to denote that a is an element of the
set A. the notation a denotes a is not an
element of the set A.

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Ways of expressing sets
• A set can be expressed in three ways:
1) Tabular Form or Roster Form
2) Descriptive Form
3) Set builder Notation

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• Tabular Form: when all the elements of set
are written within the curly bracket ‘{}’ and
elements are separated by using commas then
it is called tabular form for example;
A= {0,1,2,3,5,7} ; B= {a,b,c,d,e,f} ;
C= {10,5,15,25,20} ; D= {Raza, Hamza,
Ayesha}

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• Descriptive Form: when all the elements of a
set are described in words form it is called
descriptive form. For example:
A = set of multiple of 5 ;
B= set of vowels in English alphabet
C= set of days of a week
D= set of the first five solar months

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• Set builder Notation: when all the elements of
a set are expressed by using mathematical
notations, stating all the properties of elements
given in a set then it is called set builder
notation. For example:
 A = {0, is expressed in a set builder notation
as: A = { E}
 B = {0,1,2,3,4,5,…,15} is expressed in a set
builder notation as: B = { W }

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Types of sets
1. Equal sets
2. Singleton sets
3. Finite sets
4. Infinite sets
5. Subset
6. Proper sets

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Types of sets
1. Equal sets: two sets are equal if they only if
they have the same elements. Therefore, if A
and B are sets, then A and B are equal if and
only if x/x A x B. we write A=B if A and B are
equal sets.
For example; The set {1,3,5} and {3,5,1} are
equal. Because they have the same elements.
If the element of set {1,3,3,3,5,5,5,5} is the
same as the set {1,3,5} and equal.

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Types of sets
2. Singleton sets: a set with one element is
called singleton set. For example: A = {2}
3. Finite sets: a set which contain specific
number of different elements is called finite
set. For example: A= {1,2,3,…,100}, B=
{2,4,6,…,50}

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Types of sets
4. Infinite sets: a set which contain
uncountable number of elements is called
infinite sets. For example: A= {1,2,3,..}
5. Subset: the set A is a subset of set B. it is
denoted as A to indicate that A is a
subset of B. For example: A= {1,2,3,5} and B=
{5,4,3,2,1} then A.

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Types of sets
6. Proper Subset: if a subset of B contain at least one
element of A than Ais said to be a proper subset of B,
and B is denoted by A B.
 For example: the set of all odd positive integers less than
10 is a subset of all positive integers less than 10, the set
of rational numbers is a subset of rational number is a
subset of the set of real numbers, the set of all computer
science major at your school is a subset of the set of all
students at your school. Mathematically; find the proper
subset of A = {1,2}
Solution: {}, {1}, {2}, {1,2}

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The size of the set
• let S be a set if there are exactly n distinct
elements in S, where n is a non negative integer, we
say that S is a finite set and that n is the cardinality
of S. the cardinality of set is denoted by |S|.
For example:
 let A be the set of odd positive integers less than 10. then
|A| = 5
 Let S be the set of letters in the English alphabet. Then |
S| = 26
 And the null set has no elements it follows | = 0

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Power of Set
• In set theory, the power set (or power set) of a set A is
defined as the set of all subset of the set A including
the set itself and the null or empty set. It is denoted
by P(A). Basically, this set is the combination of all
subset including null set of a given set. For Example:
Find the power set A = { a, b, c }.
• Solution: the subset of the set are; {}, {a}, {b}, {c}, { a,
b }, { b, c }, { c, a }, {a, b, c}
• The power set; {{}, {a}, {b}, {c}, { a, b }, { b, c }, { c, a },
{a, b, c}}

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Cartesian Products
• The order of elements in a collection is often
important. Because sets are unordered, a
different structure is needed to represent
ordered collections. This is provided by
ordered n-tuples.
• Let A and B be sets, the cartesian product of A
and B, denoted by AxB, is the set of all ordered
pairs (a, b), where a A and b B. Hence, AxB =
{ (a, b) | a A b B}.

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• What is the cartesian product of A = {1, 2}
and B = { a, b , c }?
• Solution:
AxB = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}.
BxA = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)}.

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Reference
• “Quantitative reasoning: Tools for Today
informed Citizen” by Bernard L. Madison, Lynn
and Arthur Steen.
• “Quantitative reasoning for the information
Age” by Bernard L. Madison, David M.
Bressoud.
• “Fundamentals of Mathematics” by Wade
Ellis.
2/14/2024

lecture 16 quantitative reasoning.pptx

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