Definition of set:
A set can be define as " collection of objects or elements or numbers is called Set but these objects/elements/numbers/ things are distinct and well defined.
مختلف اور واضح اشیاء کے اجتماع یا اکٹھ کو سیٹ کہتے ہیں۔
Examples of set:
Dinner set,Water set,Tea set,Geometry set,Set of flowers,Set of natural numbers,Set of prime numbers, etc
The idea of set is given by a mathematician whose name is George Cantor جارج کینٹر who was born in Russia on 3rd March 1848. He died in Germany on 6 Jan. 1918. He published an article in 1873 which give birth of set theory.
Set Notations:( سیٹ لکھنے کا طریقہ)
For recognition every thing is given or represented by a name like Hydrogen is denoted by H, Oxygen is by O similarly Set is also represented by English alphabets (us captital letter to give the name of a set) and the elements or things of a set is written within the curly bracket and these elements of a set is separated by commas.
Examples:
Set of Natural numbers can be written as
N={1,2,3,4,......}
Set of whole Numbers can be written as
W={0,1,2,3,4,5,.........}
Set of pet animals can be written as
P={cow,goat,horse,sheep}
Set of flowers can be written as
F={rose, Lilly,gladia, jasmine}
Now we discuss some important set :
N= set of natural numbers ={1,2,3,4.........}
Natural numbers are start from 1 and continue......
W=set of whole Numbers ={0,1,2,3,4.......}
If we add 0 before natural numbers then it becomes whole Numbers.
E=set of even numbers ={ 0,2,4,6,8,10,.....}
The numbers which are divisible by 2 are called even numbers.
O=set of odd numbers ={1,3,5,7,9,......}
The numbers which are not divisible by 2.
P=set of prime Numbers ={2,3,5,7,11,13,....}
The numbers which have two factors, Ist is 1 and other is itself
C= set of compound numbers ={4,6,8,9,10,12,....}
The numbers which have more then two factors .
Z=set of integers ={.....-3,-2,-1,0,1,2,3,4,5.......}
Q=set of rational numbers ={ p/q , p,q ∈ Z ∧q not equal to zero}
Expressing a Set:
(سیٹ کو ظاہر کرنے کا طریقہ)
There are 3 ways to expressing a Set.
1- Descriptive form
2- Tabular form
3- Set builder form
Descriptive form:
(بیانیہ طریقہ)
If we describe a set in words OR describe with the help of a statement then it is called descriptive form.
اگر سیٹ کو اس کی خاصیت کے مطابق الفاظ میں لکھا جائے تو اس کو بیانیہ طریقہ کہتے ہیں۔
Examples:
P= set of solar planets.
S= Set of colours of rainbow
P= Set of prime Numbers
Z= Set of integers
W= Set of whole Numbers
N= set of natural numbers
F= Set of four vegetables
D= Set of five games
H= Set of pet animals , etc
Tabular Form :
(اندرانی طریقہ)
If we write the numbers or elements of a set in curly brackets ( braces) and each number/element is separate by comma ( ,) then is known as Tabular form,
Tabular Form is also known as Roster Form.
اگر کسی سیٹ کے ارکان کو {} میں "٫" کی مدد سے الگ الگ کر کے لکھا جائے تو اس کو اندرانی طریقہ کہتے ہیں۔
Examples:
Set of natural numbers can be written in Tabular Form as
N ={1,2,3,4,5...........}
Set of whole Numbers can be written in Tabular/Roster form as
W ={0,1,2,3,4,5.........}
Set of english alphabets can be written in Tabular/Roster form as
X ={ a,b,c,d,...........,x,y,z}
Set Builder Form:
(ترقیم سیٹ ساز)
If we write a set by using common property of all it's members/elements then it is called set builder form.
اگر کسی سیٹ کے تمام ارکان کو ان کی مشترکہ خصوصیت کی بنیاد پر لکھا جائے تو اس طریقہ کو ترقیم سیٹ ساز کہتے ہیں۔
Examples:
Set of natural numbers can be written in set builder form as
N={ x/x ∈ N }
Set of whole numbers can be written in set builder form as
W = {x/x ∈ W }
Set of english alphabets can be written in set builder form as
X={ x/x ∈ English alphabets }
To write a set into Set Builder Form you should know some important symbols:
< less than
≤ less than or equal to
> greater than
≥ greater than or equal to
( if the open side of symbol is on right hand then it is LESS THAN and if the open side of the symbol is on left hand then it is known as GREATER THAN)
I such that
∈ belongs to
∧ and
∨ or
Types of Sets
1- Finite sets:
متناہی سیٹ
A set which have countable / finite number of elements is called finite set.
ایسا سیٹ جس کے ارکان کی تعداد محدود ہو
Examples of finite set:
S ={ 1,2,3,4,5}
W ={ 2,6,8,9,0,1}
A= A set of first five natural numbers
C = The set of vowels
H = students of seventh class
2- Infinite Set:
غیر متناہی سیٹ
A set which have uncountable/unlimited/infinite number of elements is called an infinite set.
ایسا سیٹ جس کے ارکان کی تعداد لا محدود ہو
Examples:
A ={1,2,3,4.............}
S= { 1,3,5,7,11,........}
K = a set of whole Numbers
M = a set on integers
F= a set of compound numbers
etc.
In a finite set we can easily tell/find the last element of the set but in infinite set we cannot tell/find the last element or number of that set.
3- Empty set:
خالی سیٹ
Empty means nothing, so empty set is that set which has no element or containing nothing.
Empty set is also known as NULL set.
Empty set is denoted by a letter ∅ ( faii) فائی which is Greek letter.
It is also represented by empty brackets like { } .
ایسا سیٹ جس میں کوئی رکن نہ ہو
Examples of empty set:
A = a set of horns of an ass.
A = a set of 200 feet tall boys.
Singleton Set :
یک رکنی سیٹ
Such a set which have only be element is called singleton Set.
ایسا سیٹ جس میں ایک ہی رکن ہو
Examples :
A ={ 1}
D ={2/3}
H= a set of whole Numbers less then 1
G= a set of prime Numbers which is also even numbers
Or G= {2}
Equal and Equivalent Sets
Equal Set:
مساوی سیٹ
Two sets are said to be equal Set if the number of both sets are equal in counting and the elements of both sets are same.
For example A={1,2,3} , B={3,2,1}, we can write A=B
In these both sets we see that each set contain three elements and these three elements are also same.
ایسا سیٹ جس کے ارکان کی تعداد بھی برابر ہو اور ارکان بھی ایک جیسے ہوں۔
Equivalent Set :
مترادف سیٹ
Two sets are said to be equivalent if the number of elements of both sets are equal in number but it is not necessary that the elements of both sets are same.
Or
Two sets are equivalent if and only if one to one correspondence can be established b/w them.
ایسا سیٹ جس کے ارکان کی تعداد برابر ہو لیکن سیٹ کے ارکان کا ایک جیسا ہونا ضروری نہیں۔
ایسا سیٹ جس میں ایک سے ایک کی مطابقت قائم ہو جائے۔
Example. A={1,2,3} ,B ={ a,b,c}
These two sets are equal in number of elements but the elements are not same.
ایک مساوی سیٹ کو ہم مترادف سیٹ کہہ سکتے ہیں لیکن ایک مترادف سیٹ ،مساوی سیٹ ہو بھی سکتا ہے اور نہیں بھی ۔
Non Equivalent Set :
غیر مترادف سیٹ
The two sets are called non equivalent if one to one correspondence cannot established between these two sets.
Example:
A={1,2,3,4} , B={a,c,d,f,h,g} These two sets are non equivalent sets.
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