Lecture Notes on Relations and Functions

Introduction

• Basic principles of relations and functions.
• Understanding the types of relations and various functions.

Importance of Dedication

• Consistency is key in learning complex mathematical concepts.
• Dedication to see the lecture from start to end is crucial.

Cartesian Product

• Cartesian Product of two sets A and B is denoted as A x B.
• Essential to understand relations defined on these sets.
• Important relations concepts include set theory basics and Cartesian product calculations.

Types of Relations

• Reflexive Relation: Every element is related to itself.
• Symmetric Relation: If a is related to b, then b is related to a.
• Transitive Relation: If a is related to b and b to c, then a must be related to c.
• Exercises and examples on recognizing different types via properties.

Functions

• Definition: Special types of relations where each input has a unique output.
• Vertical Line Test: Used to determine if a graph represents a function.

Types of Functions

• Polynomial Functions: Continuous graphs, significant in calculus.
  • Odd Degree Polynomials: Range covers all real numbers.
  • Even Degree Polynomials: Range does not cover all real numbers.
• Exponential Functions: Functions in the form of f(x) = a^x, important in various growth models.
• Logarithmic Functions: Inverse of exponential functions, important for solving exponential equations.

Cartesian and Inverse Functions

• Inverse Functions: Reverse the roles of inputs and outputs.
• Only bijective functions (both one-to-one and onto) have inverses.

Periodic Functions

• Function values repeat over regular intervals.
• Examples include trigonometric functions like sine and cosine.

Injective, Surjective, and Bijective Functions

• Injective (One-to-One): Different inputs give different outputs.
• Surjective (Onto): Every possible output is covered by some input.
• Bijective: Both one-to-one and onto; has an inverse.

Odd and Even Functions

• Odd Functions: Symmetric about the origin. f(-x) = -f(x)
• Even Functions: Symmetric about the y-axis. f(-x) = f(x)

Practical Problems and Exercises

• Solving various mathematical problems to reinforce learned concepts.

Conclusion

• Summary of the importance of understanding relations and functions in higher mathematics.

Stay dedicated and focused for mastery!