## 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’ ∪ A =U A ∩ A’ = ∅ Idempotent Law And Law of null and universal set : For any finite set A 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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