Class 11 Maths Chapter 1 under the current NCERT math curricula is Set Theory and Venn Diagrams. In this chapter, you will learn what sets are, how to represent sets, and the concepts of an empty set, universal set, power set, equal sets, finite and infinite sets. Towards the end of NCERT Class 11 Maths Chapter 1, you will also learn about complement sets and solve word problems with Sets and Venn diagrams.

 

In Class 11, maths Sets can be defined as an organized collection of objects. Each set is usually represented in capitals and can be represented in two forms - the set-builder form and the roster form. 

 

A = {1,2,3,4} is a set. 

1 ∈ A, 2 ∈ A, 3 ∈ A 4 ∈ A.

 

In common mnemonics,

 

• • N: Set of all natural numbers

  • Q: Set of all rational numbers

  • R: Set of all real numbers

  • U: Universal Set

  • Z: Set of all integers

  • Z+: Set of all positive integers

 

A set that has a cardinal number equal to zero is an empty set. In other words, if a set does not contain any elements, it is known as a null set or an empty set. Any empty set is denoted by { } or Ø symbols.

Example: A set containing all the oranges in a gift basket containing only apples and pomegranates.

 

A Finite set is a type of set which consists of a definite number of elements. 

Example: A set 'X' of positive integers up to 6.

X = {1,2,3,4,5,6}

 

An Infinite set is a type of set which does not contain a defined number of elements. In other words, a set that contains an almost infinite number of elements is termed an infinite set.

Example: A set of all natural numbers from 101 onwards.

A = {101,102,103,104,105,106,107,...until infinity}

 

In Set theory, Equal sets are defined as two or more sets with exactly the same elements and cardinal number. Moreover, the order in which the elements appear in the set does not matter.

Example: A = {1,2,3,4} and B = {4,3,2,1}

A = B ( A and B are equal sets)

 

A set 'X' is said to be a subset of set Y if every element of set X is also an element of Y. This relation is denoted mathematically as A ⊆ B, where ⊆ is the symbol for a subset. Thus, a subset is a part of another set. It may also comprise a null set and the set itself.

Example: X = {1,2,3}

Then {1,2} ⊆ X.

Similarly, other subsets of set X are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.

 

A power set is defined as a set that includes all the subsets, including the empty set and the original set in itself. 

For instance, if set J = {x, y, z} is a finite set, then all its composite subsets include {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {}. These are the elements of power set, such as:

Power set of J, P(J) = { {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}, {} }. Here P(J) denotes the power set of set J.

 

A Universal set can be defined as a set that contains all the sets relevant to a certain given mathematical. In other words, a universal set is the set of all possible values under a given condition. 

Example: If X = {1,2,3} and Y {2,3,4,5}, the resultant universal set denoted by ‘U’ will be:

U = {1,2,3,4,5}

 

Venn diagrams can be defined as diagrams used to visually represent Sets in math and define the relation shared between the given sets. You can also use Venn diagrams to perform set operations in a pictorial way. 

For instance, a Venn diagram can be used to logically show different set operations such as the intersection of sets, the union of sets, a difference of sets, subsets, etc.

In set theory, several operations can be carried out when two or more sets combine to form a single set under some preexisting conditions. The following are the basic operations on sets in NCERT Class 11 Chapter 1 maths sets:

• • Union of sets: 

If we consider set P and set Q as the two sets, then P union Q is the resultant set that will contain all the elements of set P as well as set Q. It is denoted as P ∪ Q or Set P union Set Q.

For instance: If Set P = {1,2,3} and Q = {4,5,6}, then P ∪ Q = {1,2,3,4,5,6}

• • Intersection of sets:

If we consider set P and set Q as the two sets, then P intersection Q is the resultant set that will contain only the common elements between Set P and Set Q. It is denoted as P ∩ Q or set P intersection set Q.

For instance: If Set P = {0, 1, 2, 3, 4} and Q = {2, 3, 4, 5, 6}, then A ∩ B = { 2, 3, 4}

If there is no common element, the resultant intersection will be a null or void set.

The complement set of any given set can be defined as a set of all elements in the universal set that are not included in the given set. For instance, if P = { 1, 2, 3, 4, 5 } and the given universal set represents positive integers in single digits, i.e. from 1 to 9, then the complement of Set P will be equal to { 6, 7, 8, 9 }.

1. There are 500 students in a coaching centre. 220 are science enthusiasts, 180 like math, and while only 40 prefer both science and math. Find the number of students who like

1. Science but not math 

2. Math but not science 

3. Either math or science

 

Solution:

Let the total number of students be U =universal set. 

Let A is the set of students who like science.

Let B is the set of students who like math.

Thus we get: n (U) =500 n(A) =220n(B) =180n and (A∩B) =40

To find the no. of students who like science but not math, we have to find A-B.

 

A = (A-B) ∪ (A∩B) 

Which implies, n (A) = n (A-B) + n (A∩B)

To find: n (A-B) = n (A) - n (A∩B), we substitute the given values to get:

n (A-B) = 220 – 40 = 180 (Ans a)

To find the no. of students who like math but not science, we have to calculate (B-A).

Similarly as above,

n (B-A) =n (B)-n (A∩B) =180 - 40 = 140 (Ans b)

To find the number of students who like 'either' math or science, we need to find n (A∪B);

We know that n (A∪B) = n (A) + n (B) – n (A∩B)

Replacing given values:

= 220 + 180 – 40

= 360 (Ans c)

1. List all the elements in set A if A = {x:x is an odd natural number}. What type of a set is Set A?

 

Solution:

Given that A={x:x is an odd natural number}

Therefore we list the odd natural numbers as follows:

1,3,5,7,11,13...

 

Hence, set A = { 1,3,5,7,11,13…} and it is an infinite set.

 

Therefore the required answer is set A = {x:x=3n,n∈N and 1≤n≤4}

 

1. Given that set A = {a, b, c, d} and set B = {c, d, e}. Find A U B and A ⋂ B and A – B.

Solution: 

A = {a, b, c, d} and B = {c, d, e}

A U B = {a, b, c, d, e} 

A ⋂ B = {c, d} and,

 A – B = {a, b}

What are the basic concepts of Class 11 Maths Chapter 1 Sets?

The basic concepts of Class 11 Maths Set Theory include Set representation, operations on sets, and Venn diagrams to show relations and operations shared between sets.

 

What are the topics covered in Chapter 1 of NCERT Solutions for Class 11 Maths?

Chapter 1 of NCERT Solutions covers all the topics and subtopics under set theory and Venn diagrams. This includes set representation in the roster and set builder forms, set union and intersections, Venn diagrams, etc.

 

What are the important theorems that come in Class 11 Maths Chapter 1 Sets?

The important theorems in the set include the properties of union, intersection, and set difference.