Sets - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter - 1
SETS

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1. Sets and their Representations
2. The Empty Set, Finite and Infinite Sets, Equal Sets
3. Subsets, Power Set, Universal Set
4. Venn Diagrams, Operations on Sets
5. Complement of a Set
6. Union and Intersection of Two Sets      

• Set: A set is a well-defined collection of objects.
• Representaiton of sets: (i) Roster or Tabular form, (ii) Rule method or set builder form.

Types of sets:

• Empty set: A set which does not contain any element is called empty set or null set or void set. It is denoted by ϕ$\varphi$ or {  }.
• Singleton set: A set, consisting of a single element, is called a singleton set.
• Finite set: A set which consists of a definite number of elements is called  finite set.
• Infinite set: A set, which is not finite, is called infinite set.
• Equivalent sets: Two finite sets A and B are equivalent, if their cardinal numbers are same, .i.e, n(A)=n(B)$n\left(\mathrm{A}\right)=n\left(\mathrm{B}\right)$.
• Equal sets: Two sets A and B are said to be equal if they have exactly the same elements.
• Subset: A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R.
• Proper set: If A ⊆$\subseteq$ B and A ≠$\ne$ B, then A is called a proper set of B, written as A ⊂$\subset$ B.
• Universal set: If all the sets under consideration are subsets of a large set U, then U is known as a universal set. And it is denoted by rectangle in Venn-Diagram.
• Power set: A power set of a set A is collection of all subsets of A. It is denoted by P(A).
• Venn-Diagram: A gepmetrical figure illustrating universal set, subsets and their operations is known as Venn-Diagram.
• Union of sets: The union of two sets A and B is the set of all those elements which are either in A or in B.
• Intersection of sets: The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.
• Disjoint sets: Two sets A and B are said to be disjoint, if A∩B=ϕ$\mathrm{A}\cap \mathrm{B}=\varphi$.
• Difference of sets: Difference of two sets i.e., set (A - B) is the set of those elements of A which do not belong to B.
• Compliment of a set: The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A. A' = U - A.
• For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′
• If A and B are finite sets such that A ∩ B = φ, then
  n (A ∪ B) = n (A) + n (B).
• If A  ∩ B ≠ φ, then
  n (A ∪ B) = n (A) + n (B) – n (A ∩ B)