Sets:Class 11 Maths NCERT Chapter 1

Key Features of NCERT Material for Class 11 Maths Chapter 1 – Sets

In this Chapter 1: Sets we will learn about sets,its types and some operations.In the next chapter 2:Relations and functions we will study about the relations between the two objects in the pair.

Quick revision notes

Set 

A set is a very much characterized assortment of items. 

Portrayal of Set 

There are two techniques for speaking to a set 

  • Roster or Tabular form In the roster form, we list all the individuals from the set inside supports { } and separate by commas. 
  • Set-builder form In the set-builder form, we list the property or properties fulfilled by all the components of the sets. 

Sorts of Sets 

  • Empty Set: A set which doesn’t contain any component is called an empty set or the void set or null set and it is indicated by {} or Φ. 
  • Singleton Set: A set comprises of a solitary component, is known as a singleton set. 
  • Finite and infinite Set: A set which comprises of a finite number of components, is known as a finite set, in any case the set is called an infinite set. 
  • Equal Sets: Two sets A and B are supposed to be equal, if each component of A is additionally a component of B or the other way around, for example two equal sets will have the very same component.
  • Equivalent Sets: Two finite sets A and B are said to be equivalent if the number of elements in both the sets are equal, i.e. n(A) = n(B)

Subset 

A set An is supposed to be a subset of set B if each component of set A has a place with set B. In images, we compose A ⊆ B, if x ∈ A ⇒ x ∈ B 

Note: 

  • Each set is o subset of itself. 
  • The empty set is a subset of each set. 
  • The all out number of subsets of a finite set containing n components is 2n. 

Intervals as Subsets of R 

Leave an and b alone two given real numbers with the end goal that a < b, at that point 

  • an open interval indicated by (a, b) is the set of real numbers {x : a < x < b}. 
  • a closed interval meant by [a, b] is the set of real numbers {x : a ≤ x ≤ b}. 
  • intervals closed toward one side and open at the others are known as semi-open or semi-closed interval and meant by (a, b] is the set of real numbers {x : a < x ≤ b} or [a, b) is a set of real numbers {x : a ≤ x < b}. 

Power Set 

The assortment of all subsets of a set An is known as the power set of A. It is meant by P(A). If the quantity of components in A for example n(A) = n, at that point the quantity of components in P(A) = 2n. 

Universal Set 

A set that contains all sets in a given setting is known as the universal set. 

Venn-Diagrams 

Venn diagrams are the diagrams, which speak to the relationship between sets. In Venn-diagrams the universal set U is spoken to by point inside a square shape and its subsets are spoken to by focuses in closed bends (normally hovers) inside the square shape. 

Operations of Sets 

Union of sets: The union of two sets An and B, signified by A ∪ B is the set of every one of those components which are either in An or in B or in both An and B. In this manner, A ∪ B = {x : x ∈ An or x ∈ B}. 

Intersection of sets: The intersection of two sets An and B, signified by A ∩ B, is the set of all components which are basic to both An and B. 

Accordingly, A ∩ B = {x : x ∈ An and x ∈ B} 

Disjoint sets:Any Two sets, say A and B, are said to be disjoint, if A ∩ B = Φ. 

Meeting or Overlapping sets: Two sets A and B are supposed to be crossing or covering if A ∩ B ≠ Φ 

Difference of sets: For any sets An and B, their difference (A – B) is characterized as a set of components, which have a place with A yet not to B. 

Subsequently, A – B = {x : x ∈ An and x ∉ B} 

likewise, B – A = {x : x ∈ B and x ∉ A} 

Complement of a set: Let U be the universal set and A will be a subset of U. At that point, the complement of An is the set of all components of U which are not the component of A. 

In this manner, A’ = U – A = {x : x ∈ U and x ∉ A} 

A few Properties of Complement of Set

  • A ∪ A’ = ∪
  • A ∩ A’ = Φ
  • ∪’ = Φ
  • Φ’ = ∪
  • (A’)’ = A

Symmetric difference of two sets: For any set An and B, their symmetric difference (A – B) ∪ (B – A) 

(A – B) ∪ (B – A) characterized as set of components which don’t have a place with both An and B. 

It is meant by A ∆ B. 

Accordingly, A ∆ B = (A – B) ∪ (B – A) = {x : x ∉ A ∩ B}. 

Important Laws of Algebra of Sets 

Idempotent Laws: For any set A, we have

  • A ∪ A = A
  • A ∩ A = A

Identity Laws: For any set A, we have

  • A ∪ Φ = A
  • A ∩ U = A

Commutative Laws: For any two sets A and B, we have

  • A ∪ B = B ∪ A
  • A ∩ B = B ∩ A

Associative Laws: For any three sets A, B and C,  the following holds 

  • A ∪ (B ∪ C) = (A ∪ B) ∪ C
  • A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributive Laws: If A, B and C Are three sets, at that point

  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De-Morgan’s Laws: If A and B are two sets ,at that point

  • (A ∪ B)’ = A’ ∩ B’
  • (A ∩ B)’ = A’ ∪ B’

Formulas to Solve Practical Problems on Union and Intersection of Two Sets

Let A, B and C be any three finite sets, at that point

  • We can write, n(A ∪ B) = n(A) + n (B) – n(A ∩ B)
  • If (A ∩ B) = Φ, then n (A ∪ B) = n(A) + n(B)
  • We have, n(A – B) = n(A) – n(A ∩ B)
  • Also, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
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