SET THEORY
Set:-A set is a well-defined distinct collection of objects.
It is denoted by capital letters A,B,C,D,...... etc.
Representations:- There are two methods
of representing a set
1.Roaster or Tabular form:-In this form we list the elements within a curly bracket and separated by commas.
eg:-A = {3, 4, 5, 6, ... , 100}
2. Set builder form:-
eg:-A = {y : y = x + 2, x ∈ N}
Types of Set
1.Empty /Null/Void set :- A set which does not
contain any element is called the empty
set or the void set
or null set.
It is denoted by { } or ⱷ (phai).
2. Finite and infinite sets:- A set which consists of a finite number of elements is
called a finite set
otherwise, the set is called an infinite set.
3. Singleton set :- A set having only one element is called Singleton set.
4. Pair set :-A set having exactly two element is called
5. Equal sets:- Two sets A and B are said to be equal(A=B), if every elements of A is also an element of B and if every element of B is also an element of A.
6.Equivalent set :- . Two sets A and B are said to be equivalent if n(A)=n(B).
Subsets:- A set A is said to be
a subset of set B if every element of A is also an
element of B.
In
symbols we write A ⊂ B if a ∈ A ⇒ a ∈ B.
Superset:- If A ⊂ B then B is called superset of A.
Note:-Notation
1. set of real
numbers is denoted by R
2.set of natural
numbers is denoted by N
3.set of integers is denoted by Z
4.set of rational
numbers is denoted by Q
5.set of irrational
numbers is denoted by T
We observe that
N ⊂ Z ⊂ Q ⊂ R and T ⊂ R, Q ⊄ T, N ⊄ T
Intervals as subsets of R Let a, b ∈ R and a < b.
Then
(a) An open interval
denoted by (a, b) is the set of real numbers {x : a <
x < b}
(b) A closed interval denoted by [a, b] is the set
of real numbers {x : a ≤ x ≤ b)
(c) A closed open interval denoted by [a, b) is the set of real numbers {x : a ≤ x < b}
(d) An open closed interval denoted by (a, b] is the set of real numbers{x : a < x ≤ b}
Power set:- The collection of all
subsets of a set A is called the power set of A.
It is denoted by
P(A).
Note:-If the number of elements in A = n , i.e., n(A) = n,
then the
number of elements in
P(A) = nth power of 2
Universal set :- The superset of all given sets is called Universal Set.
It is denoted by U.
Operations on sets
1.Union of Sets : The union of any two given
sets A and B is the set which consists
of all those elements
which are either in A or in B.
It is denoted by A ∪ B
In symbols, we write A ∪ B = {x | x ∈A or x ∈B}
Venn Diagram
Some properties
(i) A ∪ B = B ∪ A
(ii) (A ∪ B) ∪ C = A ∪ (B ∪ C)
(iii) A ∪ ⱷ = A
(iv) A ∪ A = A
(v) U ∪ A = U
2.Intersection of sets: The intersection of
two sets A and B is the set which
consists of all those
elements which belong to both A and B.
It is denoted by A ∩ B
In symbols A ∩ B = {x : x ∈ A and x ∈ B}.
Venn diagram
Some properties
(i) A ∩ B = B ∩ A
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)
(iii) ⱷ∩ A = ⱷ
(iv) U ∩ A = A
(v) A ∩ A = A
(vi) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(vii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
3.Disjoint Sets:- When A ∩ B =ⱷ, then A and B are called disjoint sets.
Venn Diagram
It is denoted by A – B
In symbols A – B = {x : x
∈ A and x ∉ B}
Also, B – A = { x :
x ∈ B and x ∉A}
Venn Diagram
5.Complement of a set:- Let U be the
universal set and A a subset of U. Then the
complement of A is
the set of all elements of U which are not the elements of A.
It is denoted by A′
In symbols, A′ = {x : x ∈ U and x ∉ A}.
Also A′ = U – A
Venn Diagram
Some properties
(i) Law of
complements:
(a) A ∪ A′ = U (b) A ∩ A′ = ⱷ
(ii) De Morgan’s law
(a) (A ∪ B)′ = A′ ∩ B′
(b) (A ∩ B)′ = A′ ∪ B′
(iii) (A′ )′ = A
(iv) U′ = ⱷ and ⱷ′ = U
5.Symmetric Difference of two sets:-The symmetric difference of two sets A and B is denoted by AΔB, is defined as
AΔB=(A-B)U(B-A)
Formulae to solve practical problems on union and
intersection of two sets
Let A, B and C be any
finite sets. Then
(a) n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
(b) If (A ∩ B) =ⱷ, then n (A ∪ B) = n (A) + n
(B)
(c) n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n (A ∩ B) – n (A ∩ C) – n (B ∩ C) + n (A ∩ B ∩ C)
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