Download Set theory is the branch of mathematical logic that studies sets

Course
sets
7
Class
mathematics
Subject
Submitted by
Sayeda Tayyaba Shahid (1409)
Tatheer Zahra (1405)
Saba Shaukat (1442)
Submitted to
Shafiq ur rehman
The Importance of Math
Why is Math so important?
Probably because it is used in so many other subjects. There are uses of mathematics in all the
"hard" sciences, such as biology, chemistry, and physics; the "soft" sciences, such as economics,
psychology, and sociology; engineering fields, such as civil, mechanical, and industrial
engineering; and technological fields such as computers, rockets, and communications. There
are even uses in the arts, such as sculpture, drawing, and music. In addition, anything which uses
a computer uses mathematics, and you probably are aware of how many things that is!
Furthermore, learning mathematics forces one to learn how to think very logically and to solve
problems using that skill. It also teaches one to be precise in thoughts and words. Math teaches
life skills. It is difficult to find any area of life that isn't touched by mathematics. We are
surrounded by math, and also surrounded by people who do know math. If you don't know
what's going on, you are at their mercy.
Wikipedia.com
Through which ICT we share our objective
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By using internet
social media
E-mail
Multimedia
computer
Objectives
 Express a set in
 Descriptive way
 Set builder form
 Tabular form
 Define intersection, union and different of two sets.
 Find
 Union of two or more sets
 Intersection of two or more sets
 Difference of two sets
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Define and identify disjoint and overlapping sets.
Define universal set and complement of a set.
Represent set through venn diagram.
Perform operations of union, intersection, difference and complement
on two sets A and B, when:
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A is subset of B,
B is subset of A,
A and B are disjoint sets,
A and B are overlapping sets, through venn diagram.
Set theory
Set theory is the branch of mathematical logic that studies sets, which are collections of objects.
Although any type of object can be collected into a set, set theory is applied most often to objects
that are relevant to mathematics. The language of set theory can be used in the definitions of
nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the
1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were
proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom
of choice, are the best-known.
Set theory is commonly employed as a foundational system for mathematics, particularly in the
form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set
theory is a branch of mathematics in its own right, with an active research community.
Contemporary research into set theory includes a diverse collection of topics, ranging from the
structure of the real number line to the study of the consistency of large cardinals.
Wikipedia.com
What is set?
A set is a well-defined collection of distinct objects. The objects that make up a set (also known
as the elements or members of a set) can be anything: numbers, people, letters of the alphabet,
other sets, and so on. Georg Cantor, the founder of set theory, gave the following definition of a
set.
1. A set is a gathering together into a whole of definite, distinct objects of our perception or
of our thought – which are called elements of the set.
2. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if
they have precisely the same elements.
Describing sets
There are two ways of describing, or specifying the members of, a set. One way is by
intentional definition, using a rule or semantic description:
A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.
The second way is by extension – that is, listing each member of the set. An extensional
definition is denoted by enclosing the list of members in curly brackets:
C = {4, 2, 1, 3}
D = {blue, white, red}.
Every element of a set must be unique; no two members may be identical. (A multiset is a
generalized concept of a set that relaxes this criterion.) All set operations preserve this
property. The order in which the elements of a set or multiset are listed is irrelevant
(unlike for a sequence or tuple). Combining these two ideas into an example
{6, 11} = {11, 6} = {11, 6, 6, 11}
because the extensional specification means merely that each of the elements listed is a
member of the set.
Membership
The key relation between sets is membership – when one set is an element of another.
If a is a member of B, this is denoted a ∈ B, while if c is not a member of B then c ∉ B.
For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 –
4 | n is an integer; and 0 ≤ n ≤ 19} defined above,
4 ∈ A and 12 ∈ F; but
9 ∉ F and green ∉ B.
Wikipedia.com
Expressing a set
There are three ways to express a set
1)
Descriptive form
2)
Tabular form
3)
Set builder form
1) Descriptive form:
IF a set is described with the help of a statement, it is called descriptive form . for
example
N= set of natural numbers
Z= set of integers
P= set of prime numbers
W=set of whole numbers
Natural Number
The term "natural number" refers either to a member of the set of
positive integers 1, 2, 3, ... or to the set of nonnegative integers 0,
1, 2, 3, ... Regrettably, there seems to be no general agreement
about whether to include 0 in the set of natural numbers. In fact,
Ribenboim (1996) states "Let P be a set of natural numbers;
whenever convenient, it may be assumed that 0 in P."
The set of natural numbers (whichever definition is adopted) is
denoted N.
Due to lack of standard terminology, the following terms and
notations are recommended in preference to "counting number,"
"natural number," and "whole number."
Set
Name
.., -2, -1, 0, 1, 2, ...
1, 2, 3, 4, ...
0, 1, 2, 3, 4, ...
Integers
positive integers
Non-negative
integers
Non-positive integers
negative integers
0, -1, -2, -3, -4, ...
-1, -2, -3, -4, ...
symbol
Z
Z-+
Z-*
Z--
Integers
An integer is a number that can be written without a fractional or
decimal component. For example, 21, 4, and −2048 are integers;
9.75, 5½, and √2 are not integers. The set of integers is a subset of
the real numbers, and consists of the natural numbers (1, 2, 3, ...),
zero (0) and the negatives of the natural numbers (−1, −2, −3, ...).
Prime number
A prime number (or a prime) is a natural number greater than 1
that has no positive divisors other than 1 and itself. A natural
number greater than 1 that is not a prime number is called a
composite number. For example, 5 is prime because only 1 and 5
evenly divide it, whereas 6 is composite because it has the divisors
2 and 3 in addition to 1 and 6
Whole number
Whole numbers may variously refer to:
 natural numbers beginning 1, 2, 3, ...; the positive integers
 natural numbers beginning 0, 1, 2, 3, ...; the non-negative
integers
 all integers ..., -3, -2, -1, 0, 1, 2, 3, ...
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2) Tabular form:
If we list all elements of a set with in the braces {} and separate each elements by
using a comma , it is called the tabular form . For example
A={a,e,i,o,u}
C={3,6,9…..99}
W={0,1,2,3…..}
3) Set builder form:
If a set is described by using a common property of all its element, it is called set
builder form.
For example
"E is a set of even of even numbers"
Some important symbols
∧
≤
≥
∈
V
|
Such that
Less than or equal to
Greater than and equal to
Belongs to
or
Such that
Mathematics 7 by Sh. M. Tariq Rafiq, Zulqarnain Ansari
Exercise
 Write the following sets in descriptive form.
1) A= {a,e,i,o,u}
A= vowels of English alphabets.
2) B= {6,7,8,9,10}
B= set of natural numbers from 6 to 10.
 Write the following sets in tabular form.
1) E= Letters of the word “hockey”.
E= {h,o,c,k,e,y}
2) T= Multiples of 5 less than 30.
T= {5,10,15,20,25}
 Write the following sets in set builder form.
1) Z= set of natural numbers.
Z= {x | x ∈ N}
2) O= set of odd number greater than 15.
O={x | x ∈ O ∧ x >15}
Operations on sets
 Union, Intersection and difference of two sets
 Union of three sets
Following are the steps to find union of three sets
Step 1 Find the union of any two sets.
Step 2 Find the union of remaining 3rd set and the set that we get as the result of the first
step.
For three sets A, B and C there union can be taken in any of the following way.
A ∪ (B∪C)
and
={1,2,3,4} ∪ [{3,4,5,6,7,8} ∪ {6,7,8,9,10}]
={1,2,3,4} ∪ {3,4,5,6,7,8,9,10}
=
Intersection of three sets
A ∩ (B∩C)
,
and
= {a, b, c, d} ∩ ({c, d, e} ∩ {c, e, f, g})
= {a, b, c, d} ∩ {c, e}
=
Difference of two sets
Difference of set means (A-B)
= {1, 3, 6,} - {1, 2, 3, 4, 5}
=
Exercise
 Find the union of the following sets

A∪B=?
,
= {1, 3, 5} ∪ {1, 2, 3, 4}
= {1, 2, 3, 4, 5}
 Find the intersection of the following sets
,
= {3, 6, 9, 12, 15} ∩ {5, 10, 15, 20}
= {15}
Disjoint and overlapping Sets

Disjoint Sets
Two sets A and B are said to be disjoint sets, if there is no common
element between them. i.e. A = {1, 2, 3} and B = {4, 5, 6} are disjoint sets
because there is no common element in set A and set B.
 Overlapping sets
Two sets A and B are called overlapping sets , if there is at least one
element common between them but none of them is a subset of other i.e. A
= {0 , 5, 10} and B = {1, 3, 5, 7} are overlapping sets because 5 is a
common element in each sets A and B.
Universal Set and Complement of a set
 Universal Set
A set which contains all the possible elements of the sets under
consideration is called the universal set. For example, the universal set of
the counting numbers means a set that contains all possible numbers that
we can use for counting. To represent such a set we use the symbol ∪ and
read it as "universal se".
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Complement of a set
Consider a set B whose universal set is U then the difference set U\B or
ͨͨͨͨͨͨ
U-B is called the complement of set B , which is donated by B' or Bͨͨͨ and
read as "B complement ".
If U = {1, 2, 3... 10} and B = {1, 3, 7, 9} then find the B'
= {1, 2, 3…..10} - {1, 3, 7, 9}
= {2, 4, 5, 6, 8, 10}
Venn diagram
A venn diagram is simply closed figure to show sets and the relationship
between difference sets.
Mathematics 7 by Sh. M. Tariq Rafiq; Zulqarnain Ansari
 Wikipedi.com
 Answer.com
 Mathematics 7 by Sh. M. Tariq
Rafiq; Zulqarnain Ansari