Maths : ch1 Sets

Overview

This lecture fully covers the Class 11 'Sets' chapter, including the definition of sets, types, their operations, Venn diagrams, power sets, and solutions to related questions.

What is a Set and Its Representation

• A set is a collection of well-defined objects.
• Sets are written in roster form (elements listed within curly brackets) and set-builder form (according to a property).
• Well-defined means the collection is the same for everyone.

Types of Sets

• Empty Set (Φ/Null/Void): contains no elements.
• Singleton Set: contains exactly one element.
• Finite Set: contains countable elements.
• Infinite Set: contains countless elements.
• Equal Sets: sets whose elements are exactly the same.
• Equivalent Sets: sets having the same number of elements, though the elements may differ.

Subsets, Supersets, and Power Sets

• If all elements of A are in B, then A ⊆ B (A is a subset of B).
• Every set is a subset of itself; the empty set is a subset of all sets.
• Proper Subsets: subsets that do not include the set itself; total subsets are 2^n, proper subsets are 2^n - 1.
• Power Set: the set of all subsets of a set (number of elements = 2^n).

Set Operations

• Union (∪): A ∪ B = all elements of A plus all elements of B (without repetition).
• Intersection (∩): A ∩ B = elements common to both sets.
• Difference: A - B = elements in A that are not in B.
• Symmetric Difference: (A - B) ∪ (B - A)
• Complement: Universal set - A

Venn Diagrams and Laws

• Sets and their operations are represented using Venn diagrams.
• Identity/Commutative/Associative/Distributive laws, De Morgan's laws (A∪B)' = A'∩B', (A∩B)' = A'∪B')

Cardinal Number and Formulas

• Cardinal Number: total number of elements in a set.
• |A∪B| = |A| + |B| - |A∩B|
• |A - B| = |A| - |A∩B|
• |A⊕B| = |A - B| + |B - A|

Problems and Applications

• Set questions are solved using Venn diagrams and formula-based calculations.
• For union/intersection/only questions involving two or three sets, formulas/diagrams are useful.
• Disjoint Sets: sets with no common elements.

Key Terms & Definitions

• Set — a collection of well-defined objects.
• Empty Set (Φ) — contains no elements.
• Subset — A ⊆ B if all elements of A are in B.
• Power Set — the set of all subsets.
• Cardinal Number — the count of total elements in a set.
• Union — A∪B, all elements of both sets.
• Intersection — A∩B, common elements of both sets.
• Difference — A - B, elements only in A.

Action Items / Next Steps

• Practice all questions related to Venn diagrams.
• Homework: solve proving questions and union/intersection problems given in the middle or end of the lecture.
• Next topic: study Relations & Functions.