Day 34: Set Theory

Hello Dear Students,
Hope you all are doing good.

Aaj hum Sets ke baare mein study karenge, sets, its operations, and many more.. 

Let's get started...

  • Set means collection basically. 
  • In mathematics, Set is a collection of elements , and the elements are well defined and well distinct. 
  • Set is denoted by capital letter such as set A, set X.
  • The elements in the set is denoted by small letters such as a,b,c,d. 
  • For an example, X={1,2,3} and Y={a,b,c,...}
  • The above X and Y are the 2 sets and the elements of the sets are in between the curly braces. 
  • Set can be represented in 2 ways-
  • 1st way is known as Tabular method or Raster method or Enumeration method. In this, set is represented as A={1,2,3}.
  • 2nd way is known as Selector method or Set Builder method or Rule method. In this, set is represented in statement form basically, for example, A={x|xw and x is divisible by 2}. 

SUBSET - Subset means 1 set se derived doosra set. Let X and Y are 2 sets, X is subset of Y if every element of X is in Y. It is denoted by XY.
For example, Let X={1,2,3} and Y={1,2,4,6,3,9}. Then it can be said that XY, because every element of X is present in Y. 
Note that every set is a subset of itself. 
Agar hum ek set ke total number of subsets find karna chahte hain then it would be 2n. For example, A={1,2,3} and find its total number of subsets. Then, we know A has 3 elements then 23 is 8, So, A has 8 total subsets. 

POWER SET - Power set is the set of all subsets of the given set. For example, if A={1,2,3}, then power set of A, which is denoted as P(A)={ {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{Φ} }. So total subsets are 23 which is 8. 
Note that, {Φ} is always a subset of every set. 

TYPES OF SETS - The basic set types are as follows-
  1. Finite Set - Finite set wo set hota hai jisme finite number of elements ho. Finite means countable and not infinity.For example, A={a,b,c,d,e}. The set A is finite as it has finite number of elements i.e. 5 elements.
  2. Infinite Set - Infinite set wo set hota hai jisme infinite number of elements ho. Infinite means not countable means infinity. For example, A={1,2,3,4,5,6,.........}, so in this set one cannot count its elements as the elements are infinite.
  3. Disjoint Set - Disjoint set 2 sets hote hai jisme koi bhi element common nahi hota hai. In other words, if intersection of 2 sets is empty set so the sets are disjoint sets. For example, X={1,2,3,4,5} and Y={9,8,7,6}. So, X and Y mein koi bhi element common nahi hai so X and Y are disjoint sets. 
  4. Singleton Set - Singleton set wo set hota hai jisme single element hi ho,means jis set mein only 1 element hi ho. For example, A={1}. 
  5. Null Set - Null set is also known as Void set and Empty set. Null/void/empty set wo set hota hai jisme koi bhi element nahi hota means no element present in the set. It is denoted by Φ(phi).  Note that, null set is the subset of every set.
Cardinality of Set - Cardinality of set means total number of elements in the set. For example, A={1,2,3,4,5}, then the Cardinality of set A is 5. It is denoted by |A| symbol. Means |A|=5. If the Cardinality of a set is n then their total number of subsets are 2n

VENN DIAGRAM - Venn diagram basically set operations, relations ko diagrammatically show karta hai easy understanding ke liye. Venn diagram mein basically curves and circles hote hain jo ki sets ko represent karte hain. 

SET OPERATIONS - There are various operations which are performed on Sets which are as follows-
  • UNION - Union operation mein hum 2 sets ke elements ko combine kar dete hain, but elements ko repeat nahi kiya jata. The union of X and Y is represented as XY. For example, Let X={1,2,3,4,7,8,9} and Y={6,7,8,4,5}, then XY= {1,2,3,4,5,6,7,8,9}. 
  • INTERSECTION - Intersection mein hum 2 sets ke common elements ko find karte hain. Isme hum combine nahi karte but wohi elements lete hain jo common hon. The intersection of X and Y is represented as XY. For example, Let X={1,2,3,4,7,8,9} and Y={6,7,8,4,5}, then XY={4,7,8}.
  • COMPLEMENT - Let X is a subset of universal set U. Then the complement of X is all other sets which does not belongs to X in U. In other words, complement of X = U-X. Complement is denoted by Ac. For example, let U={1,2,3,4,5,6,7,8,9} and X={1,2,3}, then the complement of X is {4,5,6,7,8,9}.
  • DIFFERENCE OF 2 SETS - 2 sets ka difference simply means jo elements doosre set mein na ho. Let A and B are 2 sets, then jo elements A mein ho but B mein na ho wo A-B hoga means difference of A and B. 
Note that, A-B and B-A is not same.
For example, 
Let A={1,2,3} and B={5,3,2,6}, then, 

  • SYMMETRIC DIFFERENCE OF 2 SETS - Symmetric difference means suppose there are 2 sets, say A and B, then symmetric difference of A and B is (A-B)(B-A). It is denoted by AB. For example, A={1,2,3} and B={5,3,2,6}, then A-B={1} and B-A={5,6} and thus AB = {1}⋃{5,6}, = {1,5,6}.
  • CARTESIAN PRODUCT - Cartesian product of 2 set means basically multiplication of 2 sets. It is denoted by ×. For example, let A={1,2,3} and B={a,b}, then Cartesian product A×B= { (1,a),(1,b),(2,a),(2,b),(3,a),(3,b) }.
          Note that Cartesian product A×≠ B×A.

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