Sunday, April 26, 2020

Math : Topic 1. Set

Def. Set : '' Well defined Collections of Objects of particular kind''. Every object is called as element of set.

Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc and the elements of a set are represented by small letters a, b, c, x, y, z,etc
e.g.  V = {a, e, i, o, u.....}.If 'a' is an element of a set V, we say that “ 'a' belongs to V” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a ∈A. If ‘b’ is not an element of a set A, we write b ∉A and read “ 'b' does not belong to A”.



Fig : Repersentation of Set



Types of Set : 



  • Empty Set: A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø.
           e.g. A = {x : x is a leap year between 2016 and 2020}


  • Singleton Set: A set which contains a single element is called singleton set. 
           e.g. A = {x : x is an even prime number }


  • Finite set: A set which consists of a definite number of elements is called finite set.
           e.g. A = {x : x is number of days in an week}; A will have 7 elements. which is finite


  • Infinite set: A set which is not finite is called infinite set.
          e.g. {x: x ∈ N and x is odd} where N denotes natural numbers.



  • Equivalent set: If the cardinal number(number of elements) of the two finite sets are equal, then it is called an equivalent set. I.e, n(A) = n(B).
           

  • Equal sets: The two sets A and B are said to be equal if they have exactly the same elements.
          e.g. Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B



  • Power Sets and Subsets: The power set of a set  is the set which consists of all the subsets of the set A. It is denoted by P(A). In general, if A   is  a  set  with n(A)  = m,  then  it  can  be  shown  that n [ P(A)] = 2^m 
  • A set ‘A’ is said to be a subset of 'B' if every element of 'A' is also an element of 'B'. Intervals are subsets of R. it denoted by "⊂" mathematically A ⊂ B if a ∈ A ⇒ a ∈ B.
          e.g. if A = { 1, 2 }, then P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}, also n[ P (A) ] = 4 = 2^2. 

  • Disjoint Sets: The two sets A and B are said to be disjoint if the set does not contain any common element. 


  • Universal Set: This is the set which is the base for every other set formed. Depending upon the context, the universal set is decided. It may be a finite or infinite set. All the other sets are the subsets of the Universal set. It is represented by U.


  • Proper set: If A ⊆ B and A ≠ B, then A is called the proper set of B and it can be written as A⊂B.
       A = {1, 2, 3}, B= {1, 2, 3, 4}, A is a proper subset of B because the element 4 is not in         the first set.
  • Venn diagram: Most  of  the  relationships  between  sets  can  be represented by means of diagrams which are known as Venn diagrams.
  • Operations and Properties of Set : There are different types of set operations. Such as union of sets, intersection of sets, difference of sets and  complement of set.
  1. Union of Sets: Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol ‘∪’ is used to denote the union. Symbolically, we write A ∪ B = { x : x ∈ A or x ∈ B  } and A ∪ B usually read as ‘A union B’.                                          

    fig 1. Shaded portion is represents the union of (i) two sets (ii) three sets
                                                                
  2. Intersection of Sets: The intersection of sets A and B is the set of all elements which  are common to  both  A and B. The symbol  ‘∩’  is used to denote the intersection. The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A ∩ B = { x: x ∈ A and x ∈ B }.  B usually  read as ‘A intersection B’.Graphically as in fig 1. 
    fig 2. Shaded portion is represents the intersection of (i)two sets(ii) three sets

  3. Difference of Sets: Difference of two sets A and B is the set of elements which are present in A but not in B. It is denoted as A-B.as in fig 3.                                                                                          
  4. Complement of Set:  It is the set  of all elements in the given universal set 'U' that are not in a set 'A'. as in fig 3.
    Fig 3. (i) Difference of two sets (ii) Complement of Set A
                         

  • Properties: 


Commutative Property :
  • A B = B A
  • A ∩ B = B ∩ A
Associative Property :
  • A ( B C) = ( A B) C
  • A ∩ ( B ∩ C) = ( A ∩ B) ∩ C
Distributive Property : We will discuss it later.
  • A ( B  ∩ C) = ( A B)  ∩ (A C)
  • A ∩ ( B C) = ( A ∩ B) ( A ∩ C)
Demorgan’s Law :
  • Law of union           : ( A B )’ = A’ ∩ B’
  • Law of intersection : ( A ∩ B )’ = A’ B’
Complement Law :
  • A’ = A’  A =U
  • A ∩ A’ = 
Idempotent Law And Law of null and universal set :
For any finite set A
  • A = A
  • A ∩ A = A
  • ’ = U
  • = U’
  • (A')' = A

DeMorgan’s  laws.The complement of the union of two sets is the  intersection  of  their  complements  and  the complement  of  the  intersection  of  two  sets  is  the union of their complements. Mathematically 
  • DeMorgan's Law of union           : ( A  B )’ = A’ ∩ B’
  • DeMorgan's Law of intersection : ( A ∩ B )’ = A’  B’
Proof: DeMorgan's Law of union 
Let P = (A U B)' and Q = A' ∩ B'
Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A U B)'
⇒ x ∉ (A U B)
⇒ x ∉ A and x ∉ B
⇒ x ∈ A' and x ∈ B'
⇒ x ∈ A' ∩ B'
⇒ x ∈ Q
Therefore, P ⊂ Q …………….. (i)
Again, let y be an arbitrary element of Q then y ∈ Q ⇒ y ∈ A' ∩ B'
⇒ y ∈ A' and y ∈ B'
⇒ y ∉ A and y ∉ B
⇒ y ∉ (A U B)
⇒ y ∈ (A U B)'
⇒ y ∈ P
Therefore, Q ⊂ P …………….. (ii)
Now combine (i) and (ii) we get; P = Q i.e. (A U B)' = A' ∩ B'
Proof of De Morgan’s law intersection: (A ∩ B)' = A' U B'
Let M = (A ∩ B)' and N = A' U B'
Let x be an arbitrary element of M then x ∈ M ⇒ x ∈ (A ∩ B)'
⇒ x ∉ (A ∩ B)
⇒ x ∉ A or x ∉ B
⇒ x ∈ A' or x ∈ B'
⇒ x ∈ A' U B'
⇒ x ∈ N
Therefore, M ⊂ N …………….. (i)
Again, let y be an arbitrary element of N then y ∈ N ⇒ y ∈ A' U B'
⇒ y ∈ A' or y ∈ B'
⇒ y ∉ A or y ∉ B
⇒ y ∉ (A ∩ B)
⇒ y ∈ (A ∩ B)'
⇒ y ∈ M
Therefore, N ⊂ M …………….. (ii)
Now combine (i) and (ii) we get; M = N i.e. (A ∩ B)' = A' U B'
Few Important Symbols for representations.
N : the set of all natural numbers
Z : the set of all integers
Q: the set of all rational numbers
R : the set of real numbers
Z+: the set of positive integers
Q+: the set of positive rational numbers, and
R+: the set of positive real numbers
Some questions based on above topics

Exercise



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Reference: Above data is collected from NCERT books, Google, Wikipedia and our knowledge 




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