
Write the following set in the set-builder form:

Important Points to Remember in Chapter -1 - Sets from NCERT Mathematics Textbook for Class 11 Solutions
A set is a well-defined and distinct collection of objects and it is denoted by capital letters like
2. Representation of Sets:
There are two methods of representing a set,
(i) Roster or Tabular form:
We list all the members of the set within braces and separate by commas.
For example:
(ii) Set-builder form:
We list the property or properties satisfied by all the elements of the sets.
For example:
3. Types of Sets:
(i) Empty Set:
A set consisting of no element is called the Empty set and is denoted by
(ii) Singleton Set:
A set consisting of a single element is called a singleton set.
(iii) Finite and infinite Set:
A set consisting of a finite number of elements is called a finite set, otherwise the set is called an infinite set.
(iv) Equal Sets:
Two sets and are equal if they have exactly the same elements.
(v) Equivalent Sets:
Two finite sets and are said to be equivalent if the number of elements are equal, i.e.
(vi) Universal Set:
The set of all elements of all related sets. It is usually denoted by the symbol .
(vii) Disjoint Sets:
Two sets which do not have any element in common are called Disjoint Sets.
4. Subset & Superset:
(i) Subset:
A set is said to be a subset of a set , if every element of is also an element of i.e, , if .
(ii) Superset:
A set is said to be the Super set of a set , if every element of is also an element of .
5. Some Important Points:
(i) Every set is a subset of itself.
(ii) The empty set is a subset of every set.
(iii) The total number of subsets of a finite set containing elements is
6. Intervals as Subsets of :
If are real numbers such that then the set
(i) Closed Interval:
and is called the closed interval
(ii) Open Interval:
and is called the open interval
(iii) Semi-open or Semi-closed Interval:
(a) and is called the semi-open or semi-closed interval
(b) and is called the semi-open or semi-closed interval
7. Power Set:
The collection of all subsets of a set is called the power set of and is denoted by .
8. Operations of Sets:
(i) Union of two sets:
The union of two sets and is the set of all those elements which are either in or in or in both and is denoted by Thus,
(ii) Intersection of two sets:
The intersection of two sets and is the set of all those elements which are common to both and and is denoted by
(iii) Difference of sets:
The difference of two sets and is the set of all those elements of which do not belong to i.e. .Similarly,
(iv) Symmetric difference of two sets:
The symmetric difference of two sets and is the set and is denoted by
(v) Complement of a set:
The complement of a subset of universal set is the set of all those elements of which are not the elements of . The complement of is or .
9. Laws of Algebra of Sets:
For any three sets and , we have
(i) and (Idempotent laws)
(ii) and (Identity laws)
(iii) and (Commutative laws)
(iv) and (Associative laws)
(v) and (Distributive laws)
(vi) and (De Morgan's laws)
10. Formulae to Solve Practical Problems on Union and Intersection of Two Sets:
If and are finite sets and be the finite universal set, then
(i)
(ii) are disjoint non-void sets
(iii) i.e.,
(iv)
(v)
(vi) Number of elements in exactly two of sets and
(vii) Number of elements in exactly one of sets and