Sets:

In the present day mathematics, the concept of set serves as a fundamental part. Almost every branch of mathematics uses this concept today. Actually, sets are used to define the concepts of relations and functions. In the study of geometry, sequences, probability, etc. we require the knowledge of sets.

Sets and their representation:

Set is a collection of well-defined object. For example collection of photographs, collection of coins, natural number, prime number etc. These collections are sets. Also, we say as the river Ganges belong to the collection of rivers of India.

Representation of Set:

• Objects, elements and members of a set are synonymous (same) terms.
• Sets are generally denoted by capital letters A, B, C, X, Y, Z, etc.
• The elements of a set are represented by small letters a, b, c, x, y, z, etc.
• If a is an element of set A, we say that “ a belongs to A” the Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’. We write a ∈ If ‘b’ is not an element of a set A, we write b ∉ A and read “b does not belong to A”.

Some of the common sets used in Mathematics:

Sl. No.	Symbol	Set
1.	N	The set of all natural numbers
2.	Z	The set of all integers
3.	Q	The set of all rational numbers
4.	R	The set of real numbers
5.	Z+	The set of positive integers
6.	Q+	The set of positive rational numbers
7.	R+	The set of positive real numbers

There are two forms of representation of any set:

1. Roaster or tabular Form: In the roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.

• In roster form, the order in which the elements are represented does not matter.
• While writing the set in roster form an element is not generally repeated.

Example: Set of odd positive integers less than 10 is represented as {1, 3, 5, 7, 9}

2. Set-builder form: In set builder form of representation every element of a set possesses a single common property which is not possessed by any other element outside the set.

Example: Set of vowel in English alphabet is represented in set builder form as
V = {x : x is a vowel in English alphabet} and is written as V = {a, e, i, o, u}

No other letter possess this property.

Different types of set:

1. Empty set: A set which does not contain any element is called the empty set or the null set or the void set.
   The empty set is denoted by the symbol φ or { }.
   Example: Let A = {x : 2 < x < 3, x is a natural number}. Then A is the empty set, because there is no natural number between 2 and 3.
2. Finite set: A set which is empty or consists of a definite number of elements. We say it finite set. The number of elements in the set is uncountable.
   Example: {x : x is alphabet} this set contain 26 element.
3. Infinite set: All the set except finite set is infinite set. The number of elements in the set is uncountable.
   Example: {x : x ∈ Natural Number} This set contain infinite no. of elements.
4. Equal Sets: If two sets A & B have exactly the same elements and we write it as A = B. Otherwise, the sets are said to be unequal and we write A ≠ B.
   Example: Let A = {1, 2, 3, 4, 5} and B = {3, 1, 5, 4, 2}. Then A = B.
   There is no change in a set if one or more elements of the set are repeated.
   For example, the sets A = {a, b, c} and B = {c, c, a, b, b} are equal.

5. Subsets: A set “A” is said to be a subset of a set “B” if every element of A is also an element of B.

• A is subset of B is expressed in symbols as A ⊂ B. The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’.
• A ⊂ B if a ∈ A ⇒ a ∈ B read as “A is a subset of B if a is an element of A implies that a is also an element of B”.
• Every set A is a subset of itself, i.e. A ⊂
• Since the empty set φ has no elements that are why φ is a subset of every set.
  Example: Let A = {1, 3, 5, 7} and B = {x : x is an odd natural number less than 8}. Then A ⊂ B and B ⊂ A and hence A = B.

6. Proper subset: Let A and B be two sets. If A ⊂ B and A ≠ B, then A is called a proper subset.
   Example: A = {a, b, c} is a proper subset of B = {a, b, c, d}.
7. Superset: Let A and B be two sets. If A ⊂ B and A ≠ B , then B is called a superset.
   Example: B = {a, b, c, d} is a superset of A = {a, b, c}.
8. Singleton set: If a set has only one element, we call it a singleton set.
   Example: A = {5} is a singleton set.
9. Subsets of set of real numbers: N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄
10. Intervals as subsets of R:

• Open Interval: Let a, b ∈ R and a < b. Then the set of real numbers {y: a < y < b} is called an open interval and is denoted by (a, b).
• Close Interval: The interval which contains the end points also is called closed interval and is denoted by [ a, b ].
  Example: [ a, b ) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b.
  ( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a.
• The number (b – a) is called the length of any of the intervals (a, b), [a, b], [a, b) or (a, b].

11. Power Set: The collection of all subsets of any set is called the power set of that set. It is denoted by P(A). In P(A), every element is a set.

• If A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2^m.
  Example: if A = { 1, 2 }, then
  P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}
  n [ P (A) ] = 4 = 2²

12. Universal Set: The whole set which covers every aspect to basic is called universal set.

• The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

Operation on sets:

1. Union of sets: The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). In symbols, we write.
   A ∪ B = { x : x ∈A or x ∈B }
   some properties of union of sets:

• A ∪ B = B ∪ A (Commutative law)
• ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law )
• A ∪ φ = A (Law of identity element, φ is the identity of ∪)
• A ∪ A = A (Idempotent law)
• U ∪ A = U (Law of U)

2. Intersection of sets: 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}

Some Properties of Operation of Intersection of sets:

• A ∩ B = B ∩ A (Commutative law).
• ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
• φ ∩ A = φ, U ∩ A = A (Law of φ and U).
• A ∩ A = A (Idempotent law)
• A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) i. e., ∩ distributes over ∪.

3. Difference of sets: The difference of the sets A and B are the elements which belong to A but not to B. Symbolically, we write A – B and read as “ A minus B”.
4. Complement of a Set: Let U be the universal set and A a subset of U. The set of all elements of U which are not the elements of A is complement of set A. Symbolically, we write A′ to denote the complement of A with respect to U. Thus,

A′ = {x : x ∈ U and x ∉ A }. Obviously A′ = U – A

• If A is a subset of the universal set U, then its complement A′ is also a subset of U.

De-Morgan’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.

Some Properties of Complement of Sets:

• Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ
• De Morgan’s law: (i) (A ∪ B) ́ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
• Law of double complementation : (A′)′ = A
• Laws of empty set and universal set φ′ = U and U′ = φ.

If A and B are finite sets such that A ∩ B = φ, then

• n (A ∪ B) = n (A) + n (B).

If A ∩ B ≠ φ, then

• n (A ∪ B) = n (A) + n (B) – n (A ∩ B) 
  

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