EASY
11th CBSE
IMPORTANT
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Write the following set in the set-builder form:
 2,4,8,16,32

Important Points to Remember in Chapter -1 - Sets from NCERT Mathematics Textbook for Class 11 Solutions

1. Set:

A set is a well-defined and distinct collection of objects and it is denoted by capital letters like A,B,C...

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: A=1, 2, 3, 4

(ii) Set-builder form:

We list the property or properties satisfied by all the elements of the sets.

For example: A=x: xN, x<5

3. Types of Sets:

(i) Empty Set:

A set consisting of no element is called the Empty set and is denoted by ϕ.

(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 A and B are equal if they have exactly the same elements.

(v) Equivalent Sets:

Two finite sets A and B are said to be equivalent if the number of elements are equal, i.e. nA=nB

(vi) Universal Set:

The set of all elements of all related sets. It is usually denoted by the symbol U.

(vii) Disjoint Sets:

Two sets which do not have any element in common are called Disjoint Sets. 

4. Subset & Superset:

(i) Subset:

A set A is said to be a subset of a set B, if every element of A is also an element of B i.eAB, if xAxB.

(ii) Superset:

A set B is said to be the Super set of a set A, if every element of A is also an element of B.

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 n elements is 2n

6. Intervals as Subsets of R:

If a,b are real numbers such that a<b, then the set

(i) Closed Interval:

x:xR and axb is called the closed interval a, b

(ii) Open Interval:

x:xR and a<x<b is called the open interval a, b

(iii) Semi-open or Semi-closed Interval:

(a) x:xR and ax<b is called the semi-open or semi-closed interval a,b.

(b) x:xR and a<xb is called the semi-open or semi-closed interval a,b.

7. Power Set:

The collection of all subsets of a set A is called the power set of A and is denoted by P(A).

8. Operations of Sets:

(i) Union of two sets:

The union of two sets A and B is the set of all those elements which are either in A or in B or in both and is denoted by AB. Thus, AB=x:xAorxB.

(ii) Intersection of two sets:

The intersection of two sets A and B is the set of all those elements which are common to both A and B and is denoted by AB. 

(iii) Difference of sets:

The difference A-B of two sets A and B is the set of all those elements of A which do not belong to B i.e. AB=x:xA  andxB..Similarly, BA=x:xB  andxA.

(iv) Symmetric difference of two sets:

The symmetric difference of two sets A and B is the set ABBA and is denoted by AΔB.

(v) Complement of a set:

The complement of a subset A of universal set U is the set of all those elements of U which are not the elements of A. The complement of A is A' or Ac.

9. Laws of Algebra of Sets:

For any three sets A,B and C, we have

(i) AA=A and AA=A (Idempotent laws)

(ii) Aϕ=A and AU=A (Identity laws)

(iii) AB=BA and AB=BA (Commutative laws)

(iv) ABC=ABC and ABC=ABC (Associative laws)

(v) ABC=ABAC and ABC=ABAC (Distributive laws)

(vi) AB'=A'B' and AB'=A'B' (De Morgan's laws)

10. Formulae to Solve Practical Problems on Union and Intersection of Two Sets:

If A,B and C are finite sets and U be the finite universal set, then

(i) nAB=nA+nBnAB

(ii) nAB=nA+nBA, B are disjoint non-void sets

(iii) nAB=nAnAB i.e., nAB+nAB=nA

(iv) nAΔB=nBA+nAB=nA+nB2nAB

(v) nABC=nA+nB+nCnABnBCnCA+nABC

(vi) Number of elements in exactly two of sets A,B and C=nAB+nBC+nCA3nABC

(vii) Number of elements in exactly one of sets A,B and C=nA+nB+nC2nAB2nBC2nAC+3nABC.