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)