Subset

 In this content we discuss the following terms:

• Subset

• Types of subset

• Proper subset

• Improper subset

• Super set

• Power set

• Disjoint sets

• Overlapping sets

• Universal set

• Difference between sets

• Complement of a set

• Union of sets

• Intersection of sets

• Commutative law in sets

• Associative law in sets

• Distributive law in set

• De Morgan's law

Subset:تحتی سیٹ 

If A and B are two sets i.e

A={1,2,3,4}   , B ={1,2,3,4,5,6}

In these sets we can easily observe that all the elements of set A is present in the set B so we said that set A is subset of set B.

Subset can be define as " If every element of a set A is also the element of set B then the set A is called the subset of  set B". And it is denoted by a symbol A ⊂ B.

اگر کسی سیٹ کے تمام ارکان کسی دیے گئے سیٹ میں موجود ہوں تو وہ اس دیئے گئے سیٹ کا تحتی سیٹ کہلاتا ھے۔

If we read this D ⊂ G

set D is subset of set G.

Note:

ہر سیٹ اپنے آپ کا بھی تحتی سیٹ ہوتا ہے۔

Every set is also a subset of itself.e.g

set G is subset of set G,  so G ⊂ G

Set D is subset of set D , so D ⊂ D

Types of Subset:

There are two types of Subset

1- Proper subset

2- Improper subset

واجب تحتی سیٹ اور غیر واجب تحتی سیٹ

Proper Subset: واجب تحتی سیٹ 

If D and F are two sets, and all the elements of set D is also the element of set F but there is at least one element of set F is not present in set D, then set D is called proper subset of set F.

اگر ایک سیٹ کے تمام ارکان کسی دوسرے سیٹ میں موجود ہوں اور  دوسرے سیٹ کے کم از کم ایک رکن ایسا ہو جو پہلے سیٹ میں موجود نہ ہو تو پہلا سیٹ دوسرے سیٹ کا واجب تحتی سیٹ ہو گا۔

It is denoted by D ⊂ F.

Example:

A={1,2,3,4,5,6}  ,  S={2,4,6}

So set S is subset of set A and written as S ⊂ A.

Improper subset: غیر واجب تحتی سیٹ 

If there are two set like H and T and both set contains same elements and equal number of elements, in other words both are equal sets OR set H is subset of set T and set T is subset of set H then we say set H is improper subset of set T and set T is improper subset of set H.

ایسے دو سیٹ جن کے تمام کے تمام ارکان ایک دوسرے میں موجود ہوں۔ یا دو مساوی سیٹ آ پس میں ایک دوسرے کے غیر واجب تحتی سیٹ ہوں گے۔

Super Set: سپر سیٹ

If A ={1,2,3,4} and  

B={1,2,3,4,5,6,7,8,9,10}

In this example set A is subset of set B and it is denoted as A ⊂ B.

But the set B is the Super Set of set A and it is denoted as B ⊃ A.

Set B contains all elements of set A and also other elements.

اگر ایک سیٹ دوسرے سیٹ کا واجب تحتی سیٹ ہے تو دوسرا سیٹ پہلے سیٹ کا سپر سیٹ کہلاتا ہے۔

Note:

1- Each set is an improper subset of itself.

2- Empty set has no proper subset of itself.

Because empty set has only one subset which is empty set and it is improper subset.

3- A singleton set has only one proper subset of itself.

4- All the subsets of a set in proper subset except itself.

Power set: پاور سیٹ

A set which contains all the possible subsets of a given set is called power set.

ایسا سیٹ جو دیے گئے سیٹ کے تمام ممکنہ تحتی سیٹوں پر مشتمل ہو

Example:

D ={ 1,2}

The Power set of D can be written as

P(D) = { {} , {1} ,{2}, {1,2} }

In is example first three subsets are proper subset and 4th is improper subset of set S.

Short cut formula to fine that a set contains who many subsets of itself:

Which is 2ⁿ ( n is number of elements the set)

For example a set which contains four elements and how many subsets of this set have, we use the formula.

2ⁿ  =  2⁴  = 2×2×2×2. = 16

Disjoint Sets: غیر مشترک سیٹ 

Two sets are said to be Disjoint if there are no common element between them.

If we take intersection between these two sets then the result is empty set.

ایسے دو سیٹ جن کا کوئی بھی رکن ایک جیسا نہ ہو غیر مشترک سیٹ کہلاتے ہیں۔

Example:

A={1,2,3}  , D ={5,7,8} in this example we see there is no common element between them so set A and set D in Disjoint set.

Overlapping Sets: متراکب سیٹ 

Two sets are said to be Overlapping Sets if there is at least one element is common between them but not all the elements are common. In other words they are not subset of each other.

ایسے دو سیٹ جن کے کم از کم ایک رکن آ پس میں ملتا ہو پر سارے ارکان ایک جیسے نہ ہوں متراکب سیٹ کہلاتے ہیں۔

Examples of Overlapping Sets:

S ={1,2,3,4}  ,H ={2,4,6,8} are overlapping Sets

F={3,5,7,8,9} , G ={8,12,14,15} are overlapping Sets

Universal Set: یونیورسل سیٹ 

Universal Set is that which contains all the possible elements of the sets under consideration.

Universal Set is denoted by U.In other words universal Set in the super Set of all the given set which are under consideration.

ایسا سیٹ جو زیر غور یا زیر بحث دیئے گے سیٹوں کے تمام ارکان پر مشتمل ہو یونیورسل سیٹ کہلاتا ہے

Example:

U={1,2,3,4,5,6,7,8,9,10,11,12}

A={1,3,5}

C={2,4,6,8}

D={3,4,5,6,7}

In this example set U in universal Set of set A,set C and set D.

Difference of two sets:دو سیٹوں کا فرق 

If S and D are two sets then S difference  D is the set which contains all the elements of set S which are not the elements of set D. It is denoted by

 S - D or S/D.

And D difference S is the set which contains all the elements of set D which are not the elements of set S. It is denoted by D - S or D/S.

Note:

S - D is not equal to D - S

Complement of a set: سیٹ کا کمپلینٹ 

If A is a set and U is its universal Set then the difference set U/A is called the complement of set A and it is denoted by A' .Set A' ( complement of A) contains all the elements of universal Set U which are not the elements of set A.

یونیورسل سیٹ سے کسی بھی سیٹ کا کا فرق اس سیٹ کا کمپلینٹ کہلاتا ہے۔

Example:

U={1,2,3,4,5,6}

D={2,4,6}. Then

D' = U/D ={ 1,3,5 }

Operations on set

Union of two sets: دو سیٹ کا یونین 

If A and B are two sets them the union of set A and set B can be represented by A B and the result is a set which contains all the elements of set A or set B or both A and B.

دو سیٹ کا یونین پہلے سیٹ کے ارکان یا دوسرے سیٹ کے ارکان یا دونوں سیٹوں کے تمام ارکان پر مشتمل ہوتا ہے۔

Example:

A ={1,2,3,4}  , B {2,4,6,7}

So A ∪ B ={1,2,3,4} ∪ { 2,4,6,7}

A ∪ B ={ 1,2,3,4,6,7}  ans.

If we find B ∪ A then

B ∪ A ={2,4,6,7} ∪  { 1,2,3,4}

B ∪ A ={1,2,3,4,6,7}

So we observe the result that

A ∪ B = B ∪ A

This is called Commutative law w.r.t union

قانون مبادلہ بلحاظ یونین کہلاتا ہے۔

Intersection of two sets: دو سیٹ کا تقاطع 

If A and B are two sets them the intersection of set A and set B can be represented by A B and the result is a set which contains all the common elements of set A and B.

دو سیٹ کا تقاطع ان دونوں سیٹوں کے مشترکہ ارکان پر مشتمل ہوتا ہے 

Example:

A ={1,2,3,4}  , B {2,4,6,7}

So A ∩ B ={1,2,3,4} ∩  { 2,4,6,7}

A ∩ B ={2,4} ans.

If we find B A then

B  ∩ A ={2,4,6,7} ∩  { 1,2,3,4}

B ∩ A ={2,4} ans.

So we observe the result

A ∩ B = B ∩ A

This is called Commutative law w.r.t Intersection.

قانون مبادلہ بلحاظ تقاطع کہلاتا ہے۔

Associative Law w.r.t Union and Intersection:

قانون تلازم بلحاظ یونین اور تقاطع ۔

If S,D snd F are three sets then the Associative law can be written as

S ∪ (D ∪ F ) = (S ∪ D) ∪ F  Associstive law w.r.t union

S ∩ ( D ∩ F) = ( S ∩ D) ∩ F Associative law w.r.t intersection

Note:

To find union / intersection of three sets first of all we find union or intersection is any two sets and then the union or intersection of the third set with the resultant set.

Distributive Laws:

قانون تقسیمی

If Z, C and V are three sets then Distributive law of union over intersection is

Z ∪ ( C ∩ V ) = ( Z ∪ C ) ∩ (Z ∪ V )

And Distributive law of intersection over union is

Z ∩ ( C ∪ V ) =  (Z ∩ C ) ∪ ( Z ∩ V )

De Morgan's Laws: ڈی مارگن کے قوانین 

If  H and K are the sub set of a universal set U then

(H ∪ K)´ =H´ ∩ K´  and  (H ∩ K )´ = H´ ∪  K´

( ' ) symbol represent the compliment of a set.

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Have a nice day.