Set class 11 notes:{simple explanation with examples,PDF and worksheet}

Hey, in this post you will get set class 11 notes in simple language with lot of examples. Here is the list of topics that are going to cover in this article.

• Operation of set:[Union, intersection, disjoint, and complement of the set].

So, let’s dive into the given topic step by step and discuss every topic for set class 11 notes completely .

DEFINITION AND TYPES OF SET

SET: Set is the collection of well-defined objects.

For example, the collection of all-natural numbers is set because it is well-defined but, the collection of all good smartphones is not set because it is not well-defined, many smartphones can be in good in different features.

Note:

1. All sets are collection but all collections need not be set, for the set, a collection must be well-defined.

2. Objects in a set are called elements.

These points are really important and basic of set class 11 notes.You should

There are two ways of representing a set

1.Roster form(tabular form)

2.Set-builder form

ROSTER FORM: When the elements of a set are separated by commas, within braces{ } is called in Roster form.

Example: The set of vowels of the English alphabet may be represented as { a,e, I,o,u }.

SET-BUILDER FORM: When the elements of the set are described by the statement within the Braces { } is called in set-builder form.

Example: The set of all prime numbers less than 10 can be represented as

P= {x:x is prime numbers less than 10}

where“:” sign read as `such that ‘.

Set class 11 notes:[TYPES OF SETS]

There are six types of set

• Empty set
• Singleton set
• Finite set
• Infinite set
• Equivalent set
• Equal set
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EMPTY SET: A set is said to be empty or null or void set if it has no element and it is denoted by Φ.In roaster form, it is denoted by { }.

example: In roaster form

(i)   {x∈R:x²=-4 }=Φ.

(ii)   {x∈R:2<x<3 }=Φ.

In set-builder form

(i) The set  is given by A={x:x ∈R and x²=-9 }

Singleton set: A set consisting of a single element is called a singleton set.

example: In roster form

(i) The set {5} is a singleton set because it has only one element.

In set-builder form

(i)The set {x:x ∈N and x²=9} is singleton set because we can put two values of x to satisfy this equation i.e 3,-3 but x should belong to a natural number.so, we have only 3 as a solution.

Finite set: A set is called finite set if it is either empty set or its element can be counted in natural numbers 1,2,3,4,……etc.

Cardinal number of a finite set: The number of elements present in finite set is called a cardinal number of the set. It is denoted by n(A).

suppose a set A={4,6,6,3,0,1,}. it is a finite set because we can count the no of elements. The number of elements in set is 6 so its cardinal number will be n(A) =6.

Infinite set: A set whose elements cannot be listed by natural number (no of the element in the set is infinite ) is called an infinite set.

Example – (i) set of all points in the plane.

(ii) set of lines can pass through a point.

(iii) set of real numbers between two natural numbers.

• Cardinal number of infinite sets is infinite i.e n(A)=∞

Equivalent sets: Two finite sets A and B are equivalent if their cardinal numbers are the same i.e no of the element in both sets should be the same.

example: Set A ={1,2,3,4,5,6}

Set B={a,b,c,d,e,f}

here, both sets have the same no of element but they are different.

Equal set: Two sets A and B are said to be equal if every element of set A belong to set B, and every element of set B belong to set A i.e The cardinal number of both sets should be equal with the same elements

example -If A={1,2,4,5,6} and B={5,6,4,2,1}. Then A =B because the cardinal number of both sets is equal with the same element.

Conclusion: Every equal set is the equivalent set but every equivalent set need not be equal.

Set class 11 notes:{Subset}

Let A and B be two sets. If every element of set A belongs to set B, then A is called a subset of B and it is denoted A⊆B, which is read as ” A is a subset of B ”.

Example Let A ={1,2,3,4,5} and B={1,2,3,4,5,6} .

Here Every element of set Abelong to set B

∴                         A⊆B

Note :(i)A is a subset of  B, we say that B contains A or B is a superset of A and we write B⊃A.

(ii) If A is not a subset of B, we write A⊄B.

Subsets are further categorized as a proper and improper subset

Proper subset: A subset is called proper subset if A is a subset of B but A≠B.Above example is a proper subset

Improper subset: A subset is called an improper subset if every element of A belong to B i.e A=B

Example: All equal sets are an example of improper subsets.

• Every set is a subset of itself.
• The empty set is a subset of every set.
• An empty set is a proper subset of any set because only one element belongs to another set not all.
• Total number of a subset of the finite set containing n elements is 2^n

Universal set

The universal set is a set that contains all the set . it is a super set of all the set and it is denoted by U.

Example : If A ={1,2,3} , B={2,4,5,6} and c={1,3,5,7} ,then universal set is given by U={1,2,3,4,5,6,7}

Power set

Let A be a set. then the collection or family of all subsets of A is called the power set of A and denoted by p(A).

since the empty set and the set, A itself are subsets of A

Example Let A={1,2,3}.then ,the subsets of A are :Φ,{1},{2},{3},{1,2},{1,3},{2,3}and {1,2,3}.Hence ,p(A)={Φ,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}but If A is an Empty set ,then p(A) has just one elementΦ i.ep(A)={Φ}.

Set class 11 notes:[Operations on sets]

Union of sets

Let A and B be two sets .the union of A and B is the set of all those elements which belong either ti A or to B or to both A and B .we shall use the notation A ⋃B(read as “A union B “) to denote the union of A and B .

thus, A⋃B={x:x∈A or x∈B}

Example Let A={1,2,4,5,6}and B={a,b,c,d}

∴             A⋃B={1,2,4,5,6,a,b,c,d}

Intersection of sets

Let a and B be two sets.The intersection of A and B is the set of all those elements that belong to both A and B.The intersection of A and B is denoted by A⋂B(read as “An intersection B”)

Example Let A={1,2,4,5,6}and B={2,4,3}

∴                  A⋂B={1,2,3,4,5,6}

The intersection of above example of union

Example Let A={1,2,4,5,6}and B={a,b,c,d}

∴                  A⋂B=Φ

Disjoint sets

Two sets A and B are said to be disjoint ,if A⋂B=Φ

Example Let A={1,2,4,5,6}and B={a,b,c,d}

A⋂B=Φ

If  A⋂B≠Φ, hen A, and B are said to be intersecting or overlapping sets.

Difference of sets

Let A and B be two sets. the difference of A and B, written as A-B, is the set of all those elements of A which do not belong to B.

Thus      A-B={x:x∈A and x∉B}

similarly, the difference of B-A is the set of all those elements of B that do not belong to A

thus  B-A={x:x∉A and x∈B}

Example If A={2,3,4,5,6,7] and B={3,5,7,9,then A-B={2,4,6}because these element are not present in B .similarly because these elements are not present in A.

Symmetric difference of two sets

Let A and B be two sets .the symmetric difference of sets A and B is the sets(A-B)⋃(B-A) and is denoted by AΔB.

we have A-B={2,4,6} and B-A={9,11,13} from above example so,

(A-B)⋃(B-A)={2,4,6,9,11,13}

Complement of  a set

Let U be the universal set and let A be a set such that A⊂U.then the complement of A with respect to U is denoted by A’ or U-A and is defined as the set of all elements of U which are not in A.

Example Let U={1,2,3,4,5,6} and A={1,2}

∴            U-A={3,4,5,6}

Set class 11 notes:[Law of algebra]

Idempotent law

Statement: For any set A, A∪A=A and A∩A=A

Example: Let set A={1,2,3},

According to Idempotent law, A∪A=A and A∩A=A.Let’s check this

A∪A={1,2,3}∪{1,2,3}={1,2,3}=A                              {According to definition of union of set}

A∩A={1,2,3}∩{1,2,3}={1,2,3}=A                              {According to definition of intersection of set }

Identity law

Statement: For any set A, A∪Φ=A and A∩U =A

Example: Let set A={1,2,3},

According to identity law, A∪Φ=A and A∩U =A

As we all familiar with about Φ and U.These are the symbol to denote Empty set and universal set

A∪Φ={1,2,3}∪Φ={1,2,3}=A

A∩U={1,2,3}∩U={1,2,3}=A

Cumulative law

Statement: For any sets A and B ,A∪B=B∪A and A∩B=B∩A

That is why we can say that union and intersections of sets are always cumulative

Associative law

Statement: If A , B, and C are three sets, then (A∪B)∪C=A∪(B∪C) and A∩(B∩C)=(A∩B)∩C

i.e union and the intersection of sets are always associative

Distributive laws

Statement: If A,B and C are three sets ,then A∪(B∩C)=(A∪B)∩(A∪C) and A∩(B∪C)=(A∩B)∪(A∩C)

i.e Union and the intersection of set are distributive over intersection and union respectively