Chapter 3: 9th Class Math Notes

Chapter 3: 9th Class Math Notes are available for students with step-by-step instructions. This chapter is about sets and functions.

This chapter requires knowledge of real numbers, including set theory fundamentals, set operations, set laws and properties, power sets, Venn diagrams, word problems, and some techniques to solve the variables.

Chapter 3: 9th Class Math Notes

Class 9 Math Notes – Chapter 3 Summary

This summary covers key concepts and solution methods from Chapter 3 of your 9th class mathematics notes:

Sets

• Set Builder Notation: Used to define sets by specifying a property that all members satisfy.
  • Example: {x∣x=n2,n∈N} represents the set of squares of natural numbers.
• Tabular Form: Listing all elements of a set.
  • Example: {2,4,6,8,10} represents even numbers between 1 and 10.
• Types of Sets:
  • Universal Set (U): The set containing all elements under consideration.
  • Subsets: A set AA is a subset of BB if all elements of AA are in BB.
  • Power Set: The set of all subsets of a set.
  • Proper Subsets: Subsets that are not equal to the original set.
• Set Operations:
  • Union (A∪B): Elements in either A or B.
  • Intersection (A∩B): Elements common to both A and B.
  • Complement (A′): Elements not in A but in the universal set.
  • Difference (A−B): Elements in A but not in B.
• Properties of Sets:
  • Commutative: A∪B=B∪A, A∩B=B∩A.
  • Associative: A∪(B∪C)=(A∪B)∪C, A∩(B∩C)=(A∩B)∩C.
  • Distributive: A∪(B∩C)=(A∪B)∩(A∪C), A∩(B∪C)=(A∩B)∪(A∩C)
  • De Morgan’s Laws: (A∪B)′=A′∩B′, (A∩B)′=A′∪B′.

Venn Diagrams

• Used to visually represent sets and their relationships.
• Helps in solving problems involving unions, intersections, and complements.

Functions

• Definition: A function f from set A to set B assigns each element of A to exactly one element of B.
• Notation: f:A→B.
• Types of Functions:
  • Injective (One-to-One): Each element of B is mapped to by at most one element of A.
  • Surjective (Onto): Every element of B is mapped to by at least one element of A.
  • Bijective: Both injective and surjective.
• Function Evaluation:
  • Example: If f(x)=3x+2, then f(2)=8.
  • Given f(x)=ax+b and values of f at specific points, you can solve for constants a and b.

Word Problems Involving Sets

• Example 1: In a class of 55 students, 34 like cricket, 30 like hockey, and all students like at least one game. Find how many like both games.
  • Solution: Use the formula n(A∪B)=n(A)+n(B)−n(A∩B).
• Example 2: In a survey of 130 people, 40 like Nihari, 65 like Biryani, and 50 like Korma. Find how many like only one food.
  • Solution: Use the principle of inclusion-exclusion.

Key Formulas and Concepts

• Union of Sets: n(A∪B)=n(A)+n(B)−n(A∩B).
• Intersection of Sets: n(A∩B) represents the number of common elements.
• Complement of a Set: n(A′)=n(U)−n(A).
• Function Evaluation: Given f(x)=ax+b, solve for a and b using given values of f(x).

Practice Problems

• Set Operations: Find A∪B, A∩B, A−B, etc.
• Function Evaluation: Given f(x)=3x+2, find f(0), f(−1), etc.
• Word Problems: Solve problems involving sets and functions in real-life scenarios.

9th Class Math Notes Full Book